by Shefali Kulkarni-Thaker - University of Toronto T-Space · 2018-07-18 · I would also like to...

Inverse treatment planning for radiofrequency ablation

by

Shefali Kulkarni-Thaker

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial EngineeringUniversity of Toronto

© Copyright 2018 by Shefali Kulkarni-Thaker

Abstract

Inverse treatment planning for radiofrequency ablation

Shefali Kulkarni-Thaker

Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2018

Radiofrequency ablation (RFA) offers localized and minimally invasive thermal ablation

of small-to-medium sized inoperable superficial tumors. In RFA, needles are inserted

into the target with image guidance and current is passed through the needles, resulting

in high temperatures and consequently target ablation. However, RFA has a high local

recurrence rate caused by incomplete ablation. We therefore develop a novel two-stage

mathematical framework for pre-operative inverse treatment planning where first, we

identify needle positions and orientations using convex and integer programming tech-

niques, referred to as needle orientation optimization (NOO), and then we determine

the optimal thermal dose delivery for full target coverage by computing simultaneous

thermal and electrostatic partial differential equations, referred to as thermal dose op-

timization (TDO). In NOO, different methodologies using geometric approximations for

needle placement with and without trajectory considerations for single, multiple, and

clustered RFA applicators are presented. Using outputs from NOO, in TDO, we perform

thermal dose analysis, using several thermal damage models, for targets and organs-at-

risk (OARs). Finally, we present scenario-based thermal damage analysis to understand

the effect of translational and rotational needle deflection on target and OAR coverage.

We test our framework on three clinical cases with four different margins, for a total of

12 targets. Our methodologies provide fast treatment plans that meet clinical guidelines,

and our deflection analysis indicates that, depending on thermal damage model used, un-

ii

certain needle placement may significantly impede target coverage, and therefore, clinical

study into causes of deflection are recommended.

iii

Acknowledgements

I would like to express sincere gratitude to my supervisor, Prof. Dionne Aleman, for

giving me the opportunity to explore my research abilities. She has been highly instru-

mental in giving me untiring guidance throughout this challenging journey. Her feedbacks

have significantly helped me improve the quality of my research and its communication.

I would also like to thank our collaborator, Dr. Aaron Fenster, for providing data as well

as important feedbacks during this work. My committee members Prof. Tim Chan, who

inspires me to be a better teacher, and Prof. Roy Kwon have also been providing valuable

feedbacks during these years and their encouragement throughout the process is note-

worthy. Finally, I would like to thank Barbara and Frank Milligan Graduate Fellowship

and Ontario Research Fund for funding this research.

In the spirit of operations research, the process towards the doctoral thesis is filled

with several local optimas, and it is your peers, who are sometimes in the same boat,

that pull you through them. I want to thank Hamid Ghaffari for encouraging me to join

the department and the lab, and my labmates Kimia Ghobadi, Curtiss Luong, Vahid

Roshanaei, Ani He, Jensen Chen, and Kevin Li for helpful discussions and keeping a

lively atmosphere. I am grateful to my colleagues, Taewoo Lee, Sarina Turner, Houra

Mahmoudzadeh, Jim Kuo, and Derya Dermitas for all the fun times. The University

of Toronto’s Operations Research Group has been a big part of my early PhD years

providing quality networking, knowledge enrichment, and life-long friendships. I want

to extend my gratitude to my UTORG group: Carly Henshaw, Anna Graber-Naidech,

Justin Boutilier, Christopher Sun, Tony Tran, Elodie Rachel Mok, and Hootan Kamran.

My decision to pursue research is attributed to my father, Anil Kulkarni, a glaciologist

and researcher himself. He has been my life-long mentor and inspiration, and I am

immensely grateful for him. My mother, Vinda Kulkarni, has been my rock throughout

who reminds me the importance of grounded feet; and my brother, Ashutosh Kulkarni,

iv

for his exceptional help and support during the tough times through these years. I must

mention my grand parents, Yashwant and Nalini Maharaj, although they will not share

this moment with me, whose blessings, memories, and love will last for several years to

come.

The highest point during these years has been the birth of my son, Rohan Thaker,

whose presence bemuses me of a life before him. Wordlessly enough, he encourages me

to be a better researcher and a better person in life. Finally, I cannot express enough

gratitude in words towards my husband, Nirav Thaker, who worked hard so that I had

the luxury of studying, and it humbles me to know about your confidence in my abilities.

v

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Inverse treatment planning for RFA . . . . . . . . . . . . . . . . . . . . . 6

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Conference presentations . . . . . . . . . . . . . . . . . . . . . . . 9

2 Needle orientation optimization 11

2.1 Geometric approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 NOO without trajectory planning . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Single and clustered needle placement . . . . . . . . . . . . . . . . 14

2.2.2 Multiple needle placement . . . . . . . . . . . . . . . . . . . . . . 17

2.3 NOO with trajectory planning . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Ellipsoid definition . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Integer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Thermal dose optimization 28

3.1 Finite difference scheme for BHTE . . . . . . . . . . . . . . . . . . . . . 31

3.2 TDO with non-linear BHTE approximation . . . . . . . . . . . . . . . . 33

3.2.1 Approximation by distance and time . . . . . . . . . . . . . . . . 34

3.2.2 Approximation by isodose line . . . . . . . . . . . . . . . . . . . . 37

vi

3.3 TDO with linear BHTE approximation . . . . . . . . . . . . . . . . . . . 40

3.4 Voltage-based TDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Needle deflection analysis 45

5 Results and discussion 49

5.1 2D results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 NOO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.2 TDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 3D results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2.1 NOO: MVCE, MVCS, and NOO-Kmeans . . . . . . . . . . . . . . 60

5.2.2 NOO: Trajectory planning . . . . . . . . . . . . . . . . . . . . . . 64

5.2.3 TDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Needle deflection results . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.1 Single needle deflection . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3.2 Multiple needle deflection . . . . . . . . . . . . . . . . . . . . . . 87

5.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Conclusions and future work 92

Bibliography 95

List of acronyms 105

vii

List of Tables

3.1 Parameter values for BHTE and ATDM . . . . . . . . . . . . . . . . . . 30

5.1 Generated 2D case studies . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Computation time results for NOO-MVCE and NOO-MVCS . . . . . . . 51

5.3 Needle results for NOO-MVCE and NOO-MVCS. θ: orientation of the

needle, r: radius of the fitted ellipse or sphere. . . . . . . . . . . . . . . . 51

5.4 Optimal parameter values for dt-approx for different needle lengths . . . 53

5.5 Conformity of BHTE approximations to actual BHTE. DSC = Dice simi-

larity coefficient, CI = classic index, PI = Paddick index . . . . . . . . . 55

5.6 Treatment quality metrics showing percentage of target and OAR voxels

receiving α ≥ 320 K and conformity of the treatment plan with respect to

the target. BHTE is obtained using iso-approx treatment time. . . . . . . 57

5.7 Description of case studies . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.8 Needle types (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.9 Numerical results for NOO. A(ξ) = Fitted volumes, c = needle center, θ

= needle orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.10 Total ellipses generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.11 Total ablations with k =∞ (RMP-B) . . . . . . . . . . . . . . . . . . . . 66

5.12 Recommended needle configurations (Algorithm 7) . . . . . . . . . . . . 75

viii

5.13 Description of deflection scenarios. rmin = minimum deflection radius

(mm); rmax = maximum deflection radius (mm); |M| = total scenarios;

|M| = total sampled scenarios; |Γ| = total deflected angles; nP = number

of processors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.14 Base case needle configuration for needle deflection analysis . . . . . . . . 84

ix

List of Figures

1.1 Microwave ablation (MWA) in porcine liver . . . . . . . . . . . . . . . . 2

1.2 Ablation needle types [41] . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 A typical single ablation needle . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Insufficient tumor ablation due to incorrect needle placement . . . . . . . 4

2.1 Clustered needles [2]. Barycenter x and needle orientation a correspond to

c and θ, respectively. Note that the r is not the same as ablation radius r. 12

2.2 Cool-tip™ RF Ablation System E Series electrodes [38] . . . . . . . . . . 13

2.3 Elliptical and spherical heat maps . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Multiple needle placement using NOO-Kmeans . . . . . . . . . . . . . . . 17

3.1 Heat maps for a 30 mm needle in a 120 mm × 120 mm domain where

voxel size is 1 mm × 1 mm and the time step is 0.1 s . . . . . . . . . . . 34

3.2 Average temperature per distance at 1 minute and 20 minutes using Eu-

clidean, Chebyshev and Mahalanobis distance metrics for 30 mm needle

exposure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Relationship of time, distance, and temperature for a 30 mm needle using

the Chebyshev distance metric. . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Isodose line equilibrium for different needles . . . . . . . . . . . . . . . . 38

3.5 Quadratic (Equation QP) and exponential (Equation EP) penalties for

targets and OARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

x

4.1 Deflection scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Likelihood of a scenario occurrence . . . . . . . . . . . . . . . . . . . . . 47

4.3 Position vector angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 Slices illustrating NOO-MVCE, NOO-MVCS, and dt-approx, iso-approx

and actual full BHTE calculations. The non-gray region inside an iso-dose

line indicates its OAR coverage. . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Heat maps for a 30 mm needle in a 120 mm × 120 mm domain where

voxel size is 1 mm × 1 mm and the time step is 0.1 s . . . . . . . . . . . 54

5.3 Chebyshev distance maps (mm) . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Runtimes for NOO models NOO-MVCE and NOO-MVCS . . . . . . . . 62

5.5 Needle positions and orientations . . . . . . . . . . . . . . . . . . . . . . 63

5.6 Ellipse create runtime (min) . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.7 Comparison of pairwise validity cuts . . . . . . . . . . . . . . . . . . . . 65

5.8 Comparison between row generation approaches with RFA-SCP model . 67

5.9 Computational runtimes with bounded k . . . . . . . . . . . . . . . . . . 68

5.10 Needle placement for single and multiple needle ablation using trajectory

planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.11 Average runtimes and maximum target temperature . . . . . . . . . . . . 70

5.12 Lesion volumes (Case 1N) . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.13 Percent target coverage (Case 1N) . . . . . . . . . . . . . . . . . . . . . . 72

5.14 Percent OAR coverage (Case 1N) . . . . . . . . . . . . . . . . . . . . . . 73

5.15 Treatment times, rounded to the closest minute, based on full coverage

(Case 1N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.16 Recommended needle configuration for BHTE damage model. The voltage

is indicated by the star size. . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.17 Lesion volumes with trajectory planning (Case 1N) . . . . . . . . . . . . 77

5.18 Percent target coverage with trajectory planning (Case 1N) . . . . . . . . 78

xi

5.19 Percent OAR coverage with trajectory planning (Case 1N) . . . . . . . . 79

5.20 Single needle ablation: Target damage across scenarios . . . . . . . . . . 85

5.21 Single needle ablation: Target v. OAR D63 damage (Case 1N) . . . . . . 86

5.22 Single needle ablation: D63 target coverage (Case 1N) . . . . . . . . . . . 86

5.23 Position vector angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.24 Multi-needle ablation: Target damage across scenarios . . . . . . . . . . 88

5.25 Multi-needle ablation: Lesion volumes and inter-needle parameters (Case

2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.26 Multi-needle ablation: Needle distance and D63 damage for translational

scenes (Case 2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.27 Multi-needle ablation: Needle angles and D63 damage for combined scenes

(Case 2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xii

Chapter 1

Introduction

Hepatocellular carcinoma is the fifth-most common diagnosed malignancy, and the third-

most frequent cause of cancer-related deaths world-wide [6, 51, 64]. Although surgical

resection is the preferred treatment choice, up to 80% of these patients cannot be op-

erated on due to tumor location or existing co-morbidities [6, 45, 48]. Focal ablation

(removal of tissue with extreme temperatures), a localized, minimally invasive treatment

option for small- to medium-sized [51] tumors, is then the treatment option. Unlike

radiation, where all the organs along the paths of the beams are exposed, ablation has

fewer side effects, shorter recovery times, out-patient delivery, and minimal organs-at-risk

(OARs) damage due to localized treatment [27]. While mathematical methods to opti-

mize treatment plans have been successful in radiation therapy treatment modalities (e.g.,

intensity modulated radiation therapy, stereotactic radiosurgery, brachytherapy), there

are few similar attempts for focal ablation. We therefore present a systematic approach

to deliver focal ablation treatments with focus on radiofrequency ablation (rfa).

Focal ablative therapy is an electrosurgical procedure where an electrosurgical unit

called an electrode or a needle is inserted in to the target, which is the tumor with a

surgical margin, percutaneously, laproscopically, or via open surgery, and high frequency

alternating current is passed through it (Figure 1.1). Due to tissue resistance, electric

1

Chapter 1. Introduction 2

Figure 1.1: Microwave ablation (MWA) in porcine liver

energy converts to thermal energy causing high tissue temperatures. Heat spreads in the

surrounding tissue until a state of thermal equilibrium is achieved and ablation radius

no longer increases. In principle, focal thermal ablation aims to heat the tissue until

it cannot retain its original structure. The cell structure is reversible when heated up

to 45C unless exposed to other agents like chemotherapy and radiation [27]. Between

temperatures of 45-60C, proteins in the cells solidify, causing tissue coagulation [21].

This cellular damage increases as the cells are exposed to high temperatures for prolonged

periods of time. Thus, tissue damage depends on temperature as well as length of heat

exposure.

There are several focal ablative therapies including RFA, microwave ablation (MWA),

high intensity focused ultrasound (HIFU), laser ablation, and cryoablation. If the source

of electric current is radiowaves (10 KHz-2.59 GHz) or microwaves (300 MHz-300 GHz),

then the focal thermal ablation is called RFA or MWA, respectively. Cryoablation uses

extreme cold temperature while HIFU and laser ablation use energy beams for tissue

ablation. We focus our treatment plans on RFA, although the framework is designed to

incorporate any ablation modality.

There are several kinds of RFA needles (Figure 1.2). For instance, an ablation needle

can be clustered, which is equipment with multiple equidistant tines, or it can be an


Figure 1.2: Ablation needle types [41]

Conducting part Insulated part

Figure 1.3: A typical single ablation needle

umbrella-shaped protracted needle. The most commonly used ablation needles are a

single needle or set of single needles called multiple needles. An RFA needle, which

may be up to 30 cm long, is composed of conducting and insulating parts (Figure 1.3).

Typically, the needle is placed so that its entire conducting part (up to 4 cm long) is

inserted into the target. Since heat is delivered only through the conducting part, the

ablation regions are well controlled. We consider treatment plans delivered through single

needles, clustered needles, and multiple needles.

Despite fewer side-effects than other cancer treatment modalities, RFA has a high

local recurrence rate [6]. Inaccurate placement of the needles is common (Figure 1.4),

resulting in failure to eradicate cancerous tissue or excessive damage to surrounding

healthy tissue, and thus CT image guidance is used to help accurately position the


(a) Incorrect needle placement (b) White spots indicate inef-fective treatment.

Figure 1.4: Insufficient tumor ablation due to incorrect needle placement

needles, although at the cost of radiation exposure that may render CT usage—and

therefore highly conformal treatments—unacceptable for some patients. New ultrasound

guidance techniques achieve similar accuracy without radiation, and therefore allow for

nuanced treatments to be planned for any patient [43]. Most RFA treatment planning

considers only the impact of the needle on the targets, as opposed to the full needle

trajectory, which may render many potential solutions undeliverable (e.g., if the needle

must not pass through bone or blood vessels to reach the desired position). Further, some

works propose simultaneous needle placement and thermal delivery but are restricted

by focal ablation modality and needle type. Therefore, we propose a mathematical

optimization approach to design RFA treatments with or without consideration of needle

trajectories as well as with the flexibility to accommodate any focal ablation modality or

needle type.

1.1 Background

Unlike radiation treatments, where dose delivered from several beams is additive, heat

delivered from multiple needles is not directly additive and must be calculated using

Pennes’ bioheat transfer equation (BHTE), a partial differential equation (PDE) [65].

BHTE requires computation of the specific absorption rate (SAR), which is obtained


by solving the Laplacian, an electrostatic PDE. Further, BHTE does not consider the

amount of time a voxel, a 3D pixel, is exposed to a temperature. Alternate thermal

damage models, e.g., the Arrhenius thermal damage model (ATDM) [31, 32, 39, 40] and

cumulative equivalent minutes at 43C (CEM43) [55], use a voxel’s temperature history,

obtained from BHTE, during the course of treatment to determine tissue thermal dam-

age. These models, although non-linear, are additive across multiple needles. Further,

computation of simultaneous PDEs, BHTE and Laplacian, is a computationally intensive

task. Thus, the development of inverse plans for ablation is mathematically as well as

computationally challenging.

Existing work on RFA inverse planning can be categorized into exact and inexact

methods. Inexact methods approximate the ablated region to a sphere or an ellipse of a

known fixed size based on the needle used [11, 42, 63, 67]. A voxel within the ablated

sphere or ellipse is considered destroyed. Thus, there is no actual dose computation

and the needle is positioned by unconstrained optimization models solved using Powell’s

[11, 63, 67] or Nelder-Mead (Downhill Simplex) [42, 63] algorithms. The objectives of

these models is typically to maximize the difference in unablated target and organs-at-risk

(OARs) volumes for single [11, 63, 67] or multiple RFA applicators [42]. Although these

methods are fast and the assumption of knowing the ablation radii a priori is plausible,

they do not consider the presence of cooling effects like blood perfusion and therefore

may result in incomplete ablation.

Exact methods [2, 14, 15, 30] compute the thermal dose received by a voxel at each

time step using BHTE. Thus, there is no prior assumption on ablation radii. The de-

cision variables in exact models are the position and the orientation of the needle with

fixed treatment time, and hence, needle positioning and thermal dose optimization are

simultaneously performed. The resulting models are non-linear, constrained by a sys-

tem of PDEs describing the electric potential of the applicator and steady-state BHTE

with [30] or without [2] consideration of risk structures (e.g., ribs), and are solved using


gradient-based optimization methods [2, 30]. Models that use the Arrhenius-based ther-

mal damage model to minimize the survival fraction of the target using steepest descent

[14, 15] have better computational tractability. Since needle positioning and thermal dose

computation happen simultaneously, these models require computation of a PDE, a com-

putationally intensive task, at every new needle position and orientation. Further, these

methods are tailored to RFA-specific PDEs (electrostatic field) and therefore cannot be

immediately applied to MWA (electromagnetic field) or HIFU (acoustic field).

A list of acceptable needle trajectories is typically proposed using heuristics [4, 36, 52,

56, 57, 59, 66] and the final selection is performed using computer-assisted visualization

where each path is rated based on a linear combination of several criteria or a pareto-

optimal front [57, 59]. Haase et al. [30] used convex functions to describe forbidden regions

and developed semi-infinite techniques for trajectory planning for single needle ablation.

While most works focus on single needle trajectory planning, heuristics for sequential

placement [36] and integer models for simultaneous placement [52, 66] of multiple needles

have been explored. Sequential techniques use integer models to first identify a minimum

number of trajectories required for target coverage followed by a minimum number of

ablations required on those trajectories, resulting in inherently suboptimal solutions [52,

66]. Path length, angle of entry, and proximity to critical structures are used to determine

acceptable trajectories.

1.2 Inverse treatment planning for RFA

In inverse planning, the target and OARs are divided into unit grids called voxels (“vol-

ume pixels”) and optimal doses are sought for these structures. This methodology has

been previously used successfully to plan cancer treatments using radiosurgery [23–26]

and intensity modulated radiation therapy [54]. In a typical radiosurgery inverse plan, a

set of isocenters, which is the intersection of radiation beams, is chosen using geometric


shape approximations [25]. An optimization model is then run to determine the optimal

time to deliver radiation dose from beams with fixed or variable angles [25, 53]. Simi-

larly, in RFA we determine the position and orientation of the electrode using geometric

shape approximations, followed by the optimal treatment time for adequate thermal dose

deposition to ablate the target while sparing the OARs [35].

Ideally, simultaneous needle placement and thermal dose computations is desirable to

obtain globally optimal solutions instead of sequential optimization, which results in sub-

optimality. However, in general, PDE-constrained optimization is often computationally

difficult for large discretized systems and presents challenges, e.g., system remeshing [7],

that are beyond the scope of this work. Further, the PDEs governing thermal dose com-

putations depend on ablation modality, e.g., electrostatic fields for RFA, electromagnetic

fields for MWA, acoustic fields for HIFU, etc., and therefore, a BHTE-constrained opti-

mization model restricts its applicability to RFA treatments. Additionally, incorporating

multiple needles, which requires binary constraints, or trajectory planning, that requires

special modelling of critical structures, is difficult as they increase model complexity.

Finally, unlike radiation treatments, OAR-sparing is not of significant importance due to

the localized nature of RFA treatments. Therefore, some loss in optimality does not af-

fect the overall treatment quality. Thus, we approach the RFA inverse planning problem

in two stages.

In the first stage, called needle orientation optimization (NOO), we use geometric

shape approximations and convex optimization models to select needle positions and

orientations for single, clustered, and multiple needle scenarios. Trajectories are incorpo-

rated by eliminating undesirable needle orientations from the feasible set in single needle

scenarios, and by a set cover variation using a row generation algorithm to ensure feasi-

bility in multiple needle scenarios. In the second stage, called thermal dose optimization

(TDO), we use the NOO solution to optimize the treatment duration and voltage of the

needles to maximize thermal damage to the target. Both BHTE and ATDM damage


models are explored. NOO and TDO are performed for different needle types and source

voltages, and the best treatment plan is identified according to target and OAR damage.

Finally, we investigate target and OAR damage under diferent needle deflection scenarios

to understand the impact of delivery uncertainty.

1.3 Contributions

Our notable contribution is development of novel convex optimization models for single

or clustered needle placement and the use of row generation based integer programming

techniques for needle trajectory planning for simultaneous multiple needle placement. For

multiple needle placement without trajectory planning, we use a heuristic-based k-means

approach. Our models are solved to optimality within few minutes since the two-stage

process eliminates the need to compute PDEs at every new needle position.

We compute thermal damage using several different damage models, which allows us

to closely examine target damage at different stages of treatment and for different source

voltages. For the first time, we calculate OAR damage, which we exploit to determine the

best needle type and source voltage. This detailed analysis enables the decision maker to

obtain a complete treatment plan for the preferred damage model. Finally, for the first

time, we demonstrate the effect of needle deflection on the thermal dose delivery.

1.3.1 Publications

1. Shefali Kulkarni-Thaker, Dionne Aleman, Aaron Fenster. Inverse planning for

radiofrequency ablation in cancer therapy using multiple damage models. Submitted

to Annals of OR.

2. Shefali Kulkarni-Thaker, Dionne Aleman, Aaron Fenster. Trajectory planning

for simultaneous needle placement in radiofrequency ablation. Work in progress.

Targeted for Computers and Operations Research


3. Shen Lin, Young Kim, Sophie Tian, Shefali Kulkarni-Thaker, Dionne Aleman,

Aaron Fenster. Effect of needle deflection on thermal dose delivery in radiofre-

quency ablation. Work in progress. Targeted for IISE Transactions on Healthcare

Systems Engineering

1.3.2 Conference presentations

This work was presented by the underlined author in following conferences:

1. Shefali Kulkarni-Thaker, Dionne Aleman, Aaron Fenster. Tractable approaches

to multiple-needle radiofrequency ablation in cancer therapy, INFORMS Annual

Meeting, Philadelphia, November 2015.


radiofrequency ablation in cancer therapy using multiple needles, World Congress

on Medical Physics and Biomedical Engineering, Toronto, June 2015.

3. Shefali Kulkarni-Thaker, Dionne Aleman, Aaron Fenster. Multiple needle ra-

diofrequency ablation in cancer therapy, CORS 57th Annual Conference, Montreal,

May 2015.

4. Shefali Kulkarni-Thaker, Dionne Aleman, Aaron Fenster. A comparison of mul-

tiple damage models in cancer therapy using radiofrequency ablation, 13th Imaging

Network Ontario Symposium, London, March 2015.


focal thermal ablation using radiofrequency ablation, INFORMS Annual Meeting,

San Francisco, November 2014.

6. Shefali Kulkarni-Thaker, Dionne Aleman, Aaron Fenster. Inverse treatment

planning for radiofrequency ablation, CORS 56th Annual Conference, Ottawa, May

2014.


7. Shefali Kulkarni-Thaker, Curtiss Luong, Dionne Aleman, Aaron Fenster. In-

verse planning for single needle radiofrequency ablation in liver cancer treatment

using approximation, 12th Imaging Network Ontario Symposium, Toronto, March

2014.

8. Shefali Kulkarni-Thaker, Curtiss Luong, Dionne Aleman, Aaron Fenster. Treat-

ment planning for radiofrequency ablation using approximation, IIE Annual Con-

ference and Expo 2014, Montreal, June 2014.

9. Shefali Kulkarni-Thaker, Dionne Aleman, Aaron Fenster. Inverse treatment

planning for single radiofrequency ablation, 4th Annual MIE symposium, Toronto,

Summer 2013.

Chapter 2

Needle orientation optimization

Although our framework is independent of needle types, we explore NOO for three kinds

of needle configurations: (i) single needle, (ii) clustered needles, and (iii) 2-3 independent

needles (called multiple needles) [38]. A clustered needle is a single device with three

parallel tines (needles) that operate simultaneously (Figure 2.1). Multiple needles are

multiple single needle devices that can be inserted randomly or parallelly. They can

either be operated simultaneously or individually to ablate a larger volume or several

smaller volumes. We assume that the needles are inserted in a random order and are

operated simultaneously.

We use geometric approximations based on ellipses and spheres to identify needle

positions and orientations. For needle placement without any trajectory considerations,

we use convex ellipse or sphere covering models for single needle placement and a k-means

clustering approach for multiple needle placement. For trajectory planning, we present a

set-cover based integer programming model for simultaneous multiple needle placement.

2.1 Geometric approximations

Vendor specifications indicate an ellipsoidal (Figure 2.2(a)) and spherical (Figure 2.2(b))

shape of thermal lesions for single and clustered RFA electrodes, respectively [38], which

11

Chapter 2. Needle orientation optimization 12

Figure 2.1: Clustered needles [2]. Barycenter x and needle orientation a correspond to cand θ, respectively. Note that the r is not the same as ablation radius r.

are confirmed by our preliminary investigations (Figure 2.3). Therefore, we use ellipse-

and sphere-based geometric approximations for the placement of single/multiple and

clustered needles, respectively.

Using a fixed space separator, multiple electrodes placed parallely and operated simul-

taneously assume a spherical lesion and can be treated as a clustered electrode. However,

when needles are inserted randomly, geometric shapes are unclear. Therefore, for the pur-

pose of NOO, in the case of k multiple needles, we treat each needle as though it were

operated independently and divide the tumor into k clusters where each cluster corre-

sponds to single, and hence elliptical, needle coverage. A similar approach was used by

Chen et al. [15]. When incorporating trajectory for multiple non-parallel needle place-

ment, we predefine several ellipses, where each ellipse corresponds to a single needle with

a valid trajectory.

2.2 NOO without trajectory planning

Assuming an elliptical or spherical shape to a thermal lesion, single and clustered needle

placement the NOO problem can be represented as fitting a smallest ellipse (minimum

volume covering ellipse, MVCE) or sphere (minimum volume covering sphere, MVCS)


(a) Ellipsoid thermal lesion (b) Spherical thermal lesion for multiple parallel nee-dles (left) and clustered electrode (right)

Figure 2.2: Cool-tip™ RF Ablation System E Series electrodes [38]

35 40 45 50 55 60 65 70 75 80 8535

40

45

50

55

60

65

70

75

80

85

320

320

(a) Ellipsoidal heat contour map (b) Spherical heat contour map

Figure 2.3: Elliptical and spherical heat maps


around the target, respectively. For multiple single needles, the target is divided into

several clusters and each cluster can be treated as a single ablation NOO problem solved

using MVCE.

Let T be the set of target structures, T ′ ⊂ T be the set of boundary target voxels,

H be the set of healthy structures, and F be the set of voxels that represent forbidden

organs (critical structures), including ribs and blood vessels, through which the needle

may not pass.

2.2.1 Single and clustered needle placement

From basic algebra, the equation of an ellipse in m dimensions with center (c1, . . . , cm)

and radii (a1, . . . , am) is given by

(x1 − c1)2

a21

+ . . .+(xm − cm)2

a2m

≤ 1 (2.1)

where x1, . . . , xm are coordinates of target voxels inside the closed ellipse. Using matrix

notation, we can rewrite Equation 2.1 as a set of points, ξ:

ξ = xj | (xj − c)>A(xj − c) ≤ 1 ∀j ∈ T (2.2)

where A ∈ Sm++, a set of m ×m symmetric positive definite matrices, is full rank, and

m is the dimensionality of the matrix A, which in our case is 3 since the target is 3D.

The eigenvalue decomposition of matrix A is given by A = Q>ΣQ, where Q ∈ Rm×m is

an orthonormal matrix representing the eigenvectors of A and Σ ∈ Rm×m is a diagonal

matrix whose entries (λ1, . . . , λm) represent the eigenvalues of A. From the Principle

Axis Theorem, the square root of the inverse of the eigenvalues represent the length of

each semi-axis of the ellipse, i.e., ai =(1/√λi), while the eigenvectors, i.e., columns

of Q, represent its orientation. The volume of an ellipse is therefore proportional to


a1 × · · · × am =√

(λ1 × · · · × λm)−1 = det(√

Σ−1)

= det(√

A−1)

, where det() is the

determinant.

For any B ∈ Sm++, the determinant is given by

det(B) =n∑j=1

aij(−1)i+j det(Cij) (2.3)

where Cij is the minor matrix obtained by dropping row i and column j from ma-

trix B. Thus, the determinant is a high-degree polynomial and therefore we perform

Cholesky decomposition on B and use the simplified convex log det() function. We de-

compose B into lower triangular matrices, B = LL> where L ∈ Rm++. Thus, log det(B) =

log(det(LL>)) = log(det(L) det(L>)) = 2 log(det(L)). Since L is a lower triangular ma-

trix, its determinant is simply the product of its diagonal entries, i.e., log(det(L)) =

log∏n

i=1 `ii =∑n

i=1 log `ii. Thus, log det(B) is concave.

Now, we formulate a convex mathematical model to find the MVCE covering a set of

finite points using a log det() function [9]:

minimizeA,c

A(ξ) = log det(√

A−1)

subject to (xj − c)>A(xj − c) ≤ 1 ∀j ∈ T ′

A 0

where enforces positive definiteness on A. This model has a convex objective function,

but a non-convex constraint. However, the constraint can be reformulated as convex by

defining M =√

A and b = Mc:

maximizeM,b

log det M (NOO-MVCE)

subject to ||Mxj − b|| ≤ 1 ∀j ∈ T ′

M 0


This problem is a maximization of a concave function with convex constraints and is

thus a convex optimization problem, which can be solved to optimality. From the

global optimal solution, M∗ and b∗, we can obtain needle orientation and position by

A = (M∗)>(M∗) and c = (M∗)−1b∗, respectively. The eigenvalue decomposition of A

will give the orientation and stretch of the ellipse as described before. The eigenvector

corresponding to the smallest eigenvalue represents the longest semi-axis of the ellipse

and hence gives the orientation of the needle. The center c corresponds to the center of

the conducting part of the needle.

We treat a clustered needle applicator (Figure 2.2(b)) as a single needle, and identify

the smallest sphere covering all the voxels by formulating a convex MVCS optimization

model with an affine objective and a second-order conic constraint [10]:

minimizer,c

r (NOO-MVCS)

subject to ||xj − c|| ≤ r ∀j ∈ T ′

where r and c are the radius and center of the sphere, respectively. The tines in the

clustered applicator are fixed, parallel, and equidistant. Therefore, the center of the

fitted sphere corresponds to the barycenter or centroid of the equilateral triangle formed

by the centers of the conducting tips of each tine in the cluster. We choose the cluster

orientation along the diameter that maximizes needle-tumor contact. If the conducting

tines overlaps non-target voxels, then we rotate the cluster in increments of 5 until we

find a better needle-tumor contact. We note that the equilateral triangle has a rotational

symmetry of 120, which means the triangle (or the needles of the cluster) maps onto

itself after 120 rotation. Thus, we explore only 24 cluster rotations in a given direction

(Figure 2.1).

We note that since covering the boundary target voxels within an ellipse or sphere

also covers the internal voxels, we can reduce the constraints in models NOO-MVCE


25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

Cluster 1

Cluster 2

Cluster centers

Fitted ellipse

Figure 2.4: Multiple needle placement using NOO-Kmeans

and NOO-MVCS by only considering the boundary target voxels, i.e., j ∈ T ′, which

we obtain with a grassfire algorithm [8]. Finally, we note that model NOO-MVCS is a

special instance of NOO-MVCE, and therefore, NOO-MVCE can be used instead of NOO-

MVCS for clustered needle placement. However, the model NOO-MVCE provides more

flexibility in terms of needle orientation, while NOO-MVCS provides faster computation

time.

2.2.2 Multiple needle placement

For multiple non-parallel k needle placement, we first divide tumor voxels into k clusters

and then identify needle orientation by fitting an ellipse around each cluster using NOO-

MVCE (Figure 2.4). The methodology is referred to as NOO-Kmeans. For a set of

k needles, we use the classical k-means clustering optimization model to identify these

clusters:


minimizeµk,rjk

∑j∈T

∑k∈K

rjk||xj − µk||22 (k-means)

subject to µk =

∑∀j∈T rjkxj∑∀j∈T rjk

∀k ∈ K (2.4)

∑k∈K

rjk = 1 ∀j ∈ T (2.5)

∑∀j∈T

rjk ≥ 1 ∀k ∈ K (2.6)

rjk ∈ 0, 1 ∀j ∈ T , k ∈ K (2.7)

where µk is the mean of cluster k and rjk ∈ 0, 1 indicates if voxel j is in cluster k. The

objective is to minimize the Euclidean distance of voxels from the center (µk), thereby

assigning a voxel to a cluster k whose center is closest of all clusters. Constraint 2.4

computes the mean of a cluster, and Constraint 2.5 assigns each target voxel to a cluster.

Constraint 2.6 ensures non-empty clusters. The k-means model is nonconvex due to the

bilinear term in Constraint 2.4 as well as in the objective function. A 0-1 SDP-relaxation

of k-means has been proposed [49], but it is intractable for our problem size where the

case sizes range from approximately 900 to 62,000 target voxels. Therefore, we solve k-

means with Lloyd’s algorithm, which iteratively assigns each voxel to the nearest cluster

while updating the centroids until convergence [37].

Convex models for single, clustered, and multiple needle placements do not consider

critical structures or needle lengths, and require a priori knowledge of the number of

ablations. These limitations can be restrictive in practice, as the liver is surrounded by

critical structures like ribs and veins. Further, although OAR sparing is not imperative

for a large organ like the liver, which has regenerative properties, the use of large needles

can create large lesion volumes and undesirable OAR damage when RFA is used for

other target sites like prostate. Hence, we propose an integer model that disallows needle

placements that interfere with critical structures and OARs. By considering a ring around

the target as an OAR we can potentially avoid unnecessarily large thermal lesions.


2.3 NOO with trajectory planning

A single ablation needle’s position and trajectory corresponds to the center and orien-

tation, respectively, of an ellipse in a 3D space. A trajectory can be invalid due to its

intersection with critical structures like bone or vessels, due to physician discomfort (e.g.,

a physician will avoid inserting needle from below a supine patient), or due to an inser-

tion angle that may cause needle to slide. We predefine all ellipses, and ellipses produced

due to invalid trajectories are rejected (Section 2.3.1). Coverage of a voxel by an ellipse

can be determined given a needle position, orientation, and its vendor-specified thermal

lesion radius. Once a set of all potential needle positions and orientations are formed,

we solve an integer model to select k needles that cover the entire target. (Section 2.3.2)

Clinically, needles are inserted sequentially, but are not necessarily removed sequen-

tially. Thus, needle paths cannot cross. Further, even if each needle is removed before

the next needle is inserted, it is undesirable to place a needle center in ablated tissue

as the ablated tissue will prevent the heat from transferring to non-ablated cells due

to thermal equilibrium, and therefore needle paths still cannot cross even in a fully se-

quential insertion process. Additionally, vendor specifications recommend a minimum

distance between the needle centers to achieve a target thermal lesion, which is obtained

clinically using a separator which enforces parallel needle placement (which we consider

to be a single clustered needle scenario), preventing the treatment flexibility that could

be gained in a multiple needle scenario; however, this minimum distance can be enforced

through constraints in an optimization approach to needle placement. Finally, clinical

studies indicate poor ablation when inter-needle angles are large [5, 50]. We therefore

develop pairwise needle orientation constraints to ensure that (1) the conducting portion

of the needles do not intersect, (2) the distance between two needle centers is ≥ ω, and

(3) the angle between two needles’ major axes is ≤ α. By ensuring that each pair of

needles satisfies these rules, all needles satisfy the rules.


2.3.1 Ellipsoid definition

Recall that a single needle can be defined as an ellipse with the ellipse center at the cen-

ter of the needle’s conducting tip and the ellipse size and shape (including the principal

axis, which is the needle itself) determined by vendor-provided ablation specifications.

To generate a set of candidate ellipses, we consider every non-boundary target voxel to

be a potential ellipse center, and for each center, the set of all valid ellipse orientations

(principal axis vectors) are potential orientations. Orientations are obtained by enumer-

ating all position vectors between the target’s geometric center voxel and the boundary

voxels:

θ =g − v

||g − v||22

where v is the 3D co-ordinate of a target voxel and g is the target centroid given by

g =1

|T |∑i∈T

vi

A valid orientation adheres to user-specified rules, e.g., entry from the top of the patient

and non-oblique entry at the patient surface; while valid orientations at an ellipse center

are those that do not cause intersection between the needle conducting tip and forbidden

voxels (OARs, bones, and veins). Since this process results in an extremely large number

of ellipses, for computational tractability, we sample the full set of candidate ellipses by

only considering every pth ellipse center and only n orientations. The n orientations are

selected uniformly from the orientations that create angles in the range [30, 150] with

the patient surface (where 90 is orthogonal to the patient surface), since larger insertion

angles correspond to easier clinical delivery. Both p and n are user-provided. Based on

experimentation for tractability, we use n = 20 and

p = min

1,⌊|T |/m/|Θ|

⌋


Algorithm 1 Define ellipse set

Require: T ← Set of target voxelsRequire: H ← Set of OAR voxelsRequire: F ← Set of forbidden voxelsRequire: Θ← Set of orientationsRequire: C ← Set of valid centersRequire: r Ellipse radiiRequire: λ Needle tip length

1: Υ← ∅2: for (c, θ) ∈ C ×Θ do3: W ← linetrace(c, θ, λ) . Needle-voxel intersection set4: if |W ∩ F| = ∅ and |W ∩H| = ∅ then5: e← (c, θ, r)6: Υ← Υ ∪ e7: end if8: end for9: return Υ

where m is the user-provided maximum number of ellipses desired, |T | is the cardinality

of all the valid centers, and |Θ| = n, where Θ is the set of all orientations. Algorithm 1

shows the steps to define ellipsoids with valid trajectories.

For each ellipse e = (c,θ, r), its target and OAR coverage is determined by translating

and rotating the target and OAR voxels around the center c and orientation θ. A voxel

v ∈ T ∪H is covered by an ellipse with radii r centered at the origin and parallel to the

coordinate axes if

v2x

r2x

+v2y

r2y

+v2z

r2z

≤ 1

Finally, we create an incidence matrix E of size |T | × |Υ| to determine if target voxel

v ∈ T is covered by ellipse ei ∈ Υ, and the cost fi of selecting ellipse ei the total OAR

voxels it covers.

2.3.2 Integer model

Let F and Υ be the set of forbidden structures and predefined valid ellipses, respectively.

The goal is to cover each target voxel at least once, with at least k and at most k ellipses


or sets, while adhering to pairwise validity constraints. This formulation is a variation of

the classic set cover problem (SCP) model, and is given by

minimizez

m∑i=1

fizi (RFA-SCP)

subject to Ez ≥ 1 (2.8)

Bz ≤ d (2.9)

k ≤m∑i=1

zi ≤ k (2.10)

zi ∈ 0, 1 ∀i = 1, . . . ,m

where m = |Υ| is the number of predefined ellipses, zi indicates if ellipse ei is chosen, E

is a |T |×|Υ| incidence matrix that indicates if target voxel j ∈ T is covered by ellipse ei,

each row of matrix B identifies one or more invalid ellipse pairs, and fi is the cost of ellipse

ei defined by its total OAR voxel coverage. Thus, the objective function determines the

total cost of selected sets (ellipses), thereby minimizing OAR coverage. Constraint 2.8

ensures that each target voxel is covered at least once by the union of selected ellipses.

Due to localized nature of RFA treatment, coverage of every single voxels is essential

as needle placements can fail to cover either internal or boundary target voxels due to

insufficient heat deposition. Constraint 2.9 eliminates selection of invalid ellipse pairs

and Constraint 2.10 bounds the minimum and maximum number of ellipses or ablations.

The pairwise validity matrix, B, consists of either pairwise no-good or group cuts.

A no-good cut is an inequality that enforces at least one binary variable to change its

value. For an ellipse pair (ei, ep), a pairwise no-good cut is given by

zi + zp ≤ 1 + bip i 6= p

where bip ∈ 0, 1 indicates if ellipses ei and ep are pairwise valid. If ellipse pair (ei, ep) is


invalid, then bip = 0, which enforces selection of either ei or ep but not both. Thus, each

pairwise cut is a no-good cut that eliminates a single ellipse pair and if all the ellipse pairs

were invalid, indicating either a single ablation or an infeasible multiple needle solution,

it would generate m!/2!(m − 2)! constraints. For m ≈ 4, 000, there are up to 3,000,000

no-good cuts (Case 3M).

To reduce the number of pairwise validity constraints and improve tractability, we

propose a variation to pairwise cuts where for each ellipse ei, we generate a single group

cut of the form

βizi +∑

p∈Bi,i 6=p

zp ≤ βi

where Bi is the set of ellipses that form invalid pairs with ellipse ei and |Bi| = βi. If

βi = m − 1, then ellipse ei does not form a valid pair with any other ellipses. Thus,

unlike pairwise cuts, each group cut eliminates βi ellipse pairs. Further, at most m cuts

are required to eliminate all invalid ellipse pairs, significantly improving tractability over

pairwise no-good cuts.

The model RFA-SCP requires a priori creation of the target coverage (E) and pairwise

validity (B) matrices, potentially resulting in a memory intensive model that may not

scale to large targets. To improve computational runtime as well as to overcome memory

limitations, we design a decomposition technique based on row generation. We first solve

the model RFA-SCP with only a subset of constraints, called the reduced master problem

(RMP). Violated constraints, obtained through a feasibility check (FC), are added to the

RMP which is then resolved. The process is continued until all constraints are satisfied.

We explore a row generation approach on target coverage as well as on pairwise validity.


Target coverage row generation: The RMP when performing row generation on

target coverage is given by:

minimizez

m∑i=1

fizi (RMP-E)

subject to Ez ≥ 1 E ⊂ E

Bz ≤ d

k ≤m∑i=1

zi ≤ k

zi ∈ 0, 1 ∀i = 1, . . . ,m

For the first iteration, we consider only boundary target voxels (i.e., E = T ′), obtained

by the grassfire algorithm. We hope that by covering the boundary of the target, we also

cover the target interior. However, it is possible that the union of selected ellipses does

not cover a subset of central target voxels. Using the RMP-E solution z, we perform a

feasibility check to ensure that all target voxels are indeed covered:

minimize 0 (FC-E)

subject to Ez ≥ 1

Each violated inequality (Ez < 1) indicates an uncovered target voxel and all such

violations are added to the matrix E in model RMP-E, which is resolved. The process is

continued until FC-E returns no cuts, at which point full target coverage is achieved. If

group cuts are used to determine the pairwise validity matrix, then B ∈ Rm×m. Further,

the boundary of the target can be up to 50% of the entire target, which significantly

decreases the number of the constraints, making the model less memory intensive and

therefore more tractable. However, this approach requires the time consuming a priori


creation of the pairwise validity matrix.

Pairwise validity row generation The RMP is given by:

minimizez

m∑i=1

fizi (RMP-B)

subject to Ez ≥ 1

Bz ≤ d B ⊂ B

k ≤m∑i=1

zi ≤ k

zi ∈ 0, 1 ∀i = 1, . . . ,m

where B ⊂ B and B = ∅ for the first iteration. The selected ellipses, z, are passed to

the feasibility check subproblem:

minimize 0 (FC-B)

subject to Bz ≤ d

where B = B \ B. The FC-B is solved algorithmically to identify invalid pairs instead

of simply indicating the presence of invalid pairs, eliminating a priori creation of B.

Pairwise cuts (Algorithm 2) or group cuts (Algorithm 3) are generated using a set of

rules and cuts are passed to the RMP-B.

To account for clinical practices, we solve model RFA-SCP with unbounded k or

bounded by a small finite number to account for patient discomfort since a high number

of ablations is not desired. If a fixed set of k needles must be used, then we set k = k = k.


Algorithm 2 Create pairwise no-good cuts

Require: Υ← Set of selected ellipsesRequire: R ← Set of rules

1: B← ∅2: for (ei, ep) ∈ Υ× Υ do3: V ← checkValidity(R, ei, ep)4: if V = 0 then5: B← B ∪ zi + zp6: end if7: end for8: d← 19: return B, d

Algorithm 3 Create group cuts

Require: Υ← Set of selected ellipsesRequire: Υ← Set of predefined ellipsesRequire: R ← Set of rules

1: B,Bi ← ∅ ∀i = 1, . . . , |Υ|2: for (ei, ep) ∈ Υ×Υ do3: V ← checkValidity(R, ei, ep)4: if V = 0 then5: Bi ← Bi ∪ ep6: end if7: end for8: B←

⋃|Υ|i=1βizi +

∑p∈Bi,i 6=p zp | βi = |Bi| and βi ≥ 1

9: d←⋃|Υ|i=1βi | βi = |Bi| and βi ≥ 1

10: return B, d

Algorithm 4 checkValidity: Pairwise validity check

Require: ei, ep ← Pair of ellipsesRequire: R ← Set of rules

1: V = 12: for r ∈ R do3: if (ei, ep) does not satisfy r then4: V = 05: break6: end if7: end for8: return V


Depending on the maximum ellipse coverage, the value of k is selected as follows:

k =

1 if max

[∑j∈T Eji

]≥ |T |

2 otherwise

The IP approaches presented here require creation of all the valid trajectories a priori,

which can have significant overhead, especially when high accuracy in needle placement

is desirable. However, ellipse creation is an embarrassingly parallel problem and runtimes

can be significantly improved. Further, the trajectory planning models assume a vendor-

specified lesion volume for each ellipse or needle, although, in practice, thermal doses

will vary from this estimation due to several factors including local tissue properties and

interactions. Additionally, a target or OAR voxel may be covered multiple times due

to overlapping ellipses but the actual dose at a voxel is not additive. Therefore, during

the TDO stage, we disregard the assumption on lesion sizes and compute true thermal

distributions (Chapter 3).

Chapter 3

Thermal dose optimization

Once needle positions are known, we lift the geometric assumptions made during NOO

and compute the actual thermal dose received. Thermal dose can be computed as a

temperature or a percentage cellular damage value. A minimum of 60C temperature

is desirable for irreversible target damage, which corresponds to 63% cellular damage.

These temperature and percentage damage values are computed using BHTE and ATDM,

respectively [31, 32, 39, 40, 65].

BHTE describes the relationship between tissue local interactions and heat delivery,

and is given by the following equation in a 3D system [1, 65]:

ρ`c`∂T

∂t= K

(∂2T

∂x2+∂2T

∂y2+∂2T

∂z2

)− cbρbω(T − Ta) +Qm +Qp (BHTE)

where ρ` and ρb are the densities of tissue and blood (kg/m3), respectively; c` and cb

are the specific heats of the tisue and blood (J/kg-K), respectively; K is the thermal

conductivity of the tissue (W/m-K); ω is the blood perfusion coefficient, i.e., blood flow

rate/unit mass tissue (1/s); T and Ta are the temperatures of tissue and arterial blood

(K), respectively; Qp is the power absorbed per unit volume of the tissue (W/m3); and

Qm is metabolic heating, which is usually considered negligible [13]. The values used for

the biological constants and other parameters are given in Table 3.1. The solution of

28

Chapter 3. Thermal dose optimization 29

BHTE gives the temperature of each voxel at each time step. We note that the presence

is large blood vessels near the needle can affect the heat flow due to the heat-sink effect,

where the cooler blood flow in the veins can reduce the effective target temperatures.

Clinically, blood occlusion may be performed to temporarily restrict the blood flow,

thereby reducing its effect on target temperatures. The BHTE does not consider the

heat loss due to these large veins. Studies that address the heat-sink effect may be

used for accurate temperature distributions [33], especially for targets located near veins,

however, we ignore heat sinks for computational simplicity.

The heat source, Qp, is approximated by [14, 15]

Qp = σ`||∇Φ||22 (SAR)

where σ` is the electrical conductivity of the tissue and Φ is the electric potential. We

obtain the electric potential using the Laplacian equation with constant electrical con-

ductivity [12] as follows:

∂2Φ

∂x2+∂2Φ

∂y2+∂2Φ

∂z2= 0 (Laplacian)

The needle is positioned so that the center of its conducting part is placed at the

ellipse or sphere center obtained from NOO. The voxels in contact with the needle are

computed from a ray tracing algorithm [3] and form a needle-voxel intersection set. For

Laplacian, the initial conditions (voltage) are set to 0 for all voxels except the needle-voxel

intersection set, whose initial conditions are set to input voltage of the needle.

The Arrhenius thermal damage index is a dimensionless number Ωjs computed for

every voxel j of structure s and may be interpreted as the probability that the tissue is

irreversibly damaged [47]. ATDM is defined as

Ωjs (t) =

∫ t

0

A exp

(−EART (t)

)dt (ATDM)


Table 3.1: Parameter values for BHTE and ATDM

Parameter Value

Blood density(ρb) [62] 1000 kg/m3

Blood heat capacity (cb) [62] 4180 J/kg-KBlood thermal conductivity [62] 0.543 W/m-KLiver density (ρ`) [62] 1060 kg/m3

Liver heat capacity (ρ`) [62] 3600 J/kg-KLiver thermal conductivity (K) [62] 0.512 W/m-KLiver electrical conductivity (σ`) [62] 3.33E-3 mS/cmBlood perfusion (ω) [20] 6.4E-3 1/sArterial temperature (Ta) 310.15 KFrequency factor (A) [58] 3.1E98 1/sActivation energy (EA) [58] 6.28E5 J/moUniversal gas constant (R) [44] 8.3145 J/K-mol

where A is the frequency factor, EA is the activation energy, and R is the universal gas

constant (Table 3.1). T (t) is the average tissue absolute temperature (i.e., temperature

in Kelvin) in the time interval [0, t] and is obtained from BHTE. Physically, Ωjs is a

natural log of the ratio of the original concentration of undamaged molecules to those at

the end of the heating [47]:

Ωjs = ln

C0

Ct

=

original concentration of undamaged molecules

undamaged molecules at time t

Thus, if Ct ∈ [0, 1] and C0 = 1, then percentage (or probability) of damaged molecules

at time t is

D = 1− exp(−Ωjs) (Percent damage)

because exp(−Ωjs) = Ct = undamaged molecules at time t. We describe these percent-

age damage models as D63 for p = 63% tissue damage, D70 for p = 70% tissue damage,

etc. A value of p = 0.63 or 63% is associated with irreversible thermal damage and

corresponds to Ωjs = 1.

The CEM43 model is the cumulative equivalent time at reference temperature 43C.


It quantifies the damage in a non-linear fashion using temporal temperature [55]:

Ψjs (t) =t∑i=0

R|43−T (ti)|CEM ∆t (CEM43)

T (ti) is the temperature of the voxel at time ti and ∆t is the time of exposure (minutes)

at that temperature. The value RCEM is the constant of proportionality and is usually

set to ≈ 0.5 at or above breakpoint temperatures (43C); for other temperatures it is set

to 0.25 [55, 61]. We note that RFA operates on high temperatures making the CEM43

model unstable, and hence not considered, for thermal damage computations.

3.1 Finite difference scheme for BHTE

We use an explicit finite difference scheme in 3D to solve the BHTE and the Laplacian

equations. The electrostatic equation given by Laplacian in 3D is as follows:

∂2Φ

∂x2≈ ∇

2Φ

∇2x=

Φ(x+ v, y, z) + Φ(x− v, y, z)− 2Φ(x, y, z)

v2

∂2Φ

∂y2≈ ∇

2Φ

∇2y=

Φ(x, y + v, z) + Φ(x, y − v, z)− 2Φ(x, y, z)

v2

∂2Φ

∂z2≈ ∇

2Φ

∇2z=

Φ(x, y, z + v) + Φ(x, y, z − v)− 2Φ(x, y, z)

v2

where Φ(x, y, z) is the voltage at position (x, y, z) and v is the dimension of the voxel.

Let Ψ represent the problem domain, and Ψb ⊂ Ψ represent the voxels at the boundary

of the domain. T ⊂ Ψ be the set of voxels representing the tumor, TN ⊂ T be the voxels

in contact with the needle. The initial conditions are

Φ(x, y, z) = 0 ∀x, y, z ∈ Ψ

Φ(x, y, z) = Φ0 ∀x, y, z ∈ TN


and the homogeneous Neumann boundary conditions [60] are

∂Φ

∂x=∂Φ

∂y=∂Φ

∂z= 0 ∀x, y, z ∈ Ψb

where Φ0 is the input voltage.

In Equation BHTE, the temperature is evaluated as follows:

∂T

∂t≈ ∇T∇t

=T (t, x, y, z)− T (t−∆t, x, y, z)

∆t(3.1)

where T (t, x, y, z) is the temperature of a voxel at position (x, y, z) at time t, and ∆t is

the time step length or the frequency in seconds when temperature measures are made.

The spatial coordinates are approximated as follows:

∂2T

∂x2≈ ∇

2T

∇2x=T (t, x+ v, y, z) + T (t, x− v, y, z)− 2T (t, x, y, z)

v2

∂2T

∂y2≈ ∇

2T

∇2y=T (t, x, y + v, z) + T (t, x, y − v, z)− 2T (t, x, y, z)

v2

∂2T

∂z2≈ ∇

2T

∇2z=T (t, x, y, z + v) + T (t, x, y, z − v)− 2T (t, x, y, z)

v2

The initial condition is

T (0, x, y, z) = Tbody ∀x, y, z ∈ Ψ (3.2)

and the homogeneous Neumann boundary conditions [60] are

∂T

∂x=∂T

∂y=∂T

∂z= 0 ∀x, y, z ∈ Ψb

where τ is total simulation time, Tbody = 310.15K is the body temperature. We use voxel

size 1 mm × 1 mm × 1 mm or v = 1 mm and the time step is ∆t = 0.5 s.


3.2 TDO with non-linear BHTE approximation

BHTE calculations are computationally expensive, and we therefore examine two ap-

proximations to BHTE for improved computational performance and tractability: (1)

an exponential function of distance and time optimized using least squares, and (2) an

isodose line growth function that is heuristically optimized. We illustrate our approach

on synthetic 2D slices using single and clustered needle ablation.

In a 2D Cartesian coordinate system, Pennes’ BHTE is

ρ`c`∂T

∂t= K

(∂2T

∂x2+∂2T

∂y2

)− cbρbω(T − Ta) +Qm +Qp (BHTE-2D)

To approximate BHTE-2D, we must understand the nature of the PDE. For prototyping,

we solve BHTE-2D first in a 2D domain of size 120 mm × 120 mm with an electrode

of 1 mm thickness with different exposure lengths placed at the center of the domain

in a direction perpendicular to the y-axis. We solve BHTE-2D using the explicit finite

difference scheme with a voxel size of 1 mm × 1 mm and simulate over 1200 s with a

time step of 0.1 s. We also solve BHTE-2D for various nedle lengths, for a cluster of

three needles, each of same length, placed 10 mm apart, and parallel to each other. The

thermal distributions over time for a single needle and a cluster of three needles with 30

mm exposure are shown in Figure 3.1.

We note that BHTE-2D reaches thermal equilibrium within few minutes, after which

the radius of the ablated zone does not increase. Based on these heat maps and equilib-

rium observations, we devise two approaches to approximate BHTE-2D. The first method,

called dt-approx, is to estimate a voxel’s thermal damage as a function of its distance

from the needle and needle exposure time. The second method, called iso-approx, ap-

proximates isodose line growth as a function of time. Each approximation method has

its own TDO approach.


SingleNeedle

ClusteredNeedles

1 min 5 min 10 min

Figure 3.1: Heat maps for a 30 mm needle in a 120 mm × 120 mm domain where voxelsize is 1 mm × 1 mm and the time step is 0.1 s

3.2.1 Approximation by distance and time

We determine the distance metric to use by performing experiments using the Cheby-

shev, Euclidean, and the Mahlanobis distance metric on a 30 mm needle exposure. We

study average temperatures, obtained using temperatures of voxels at same distance,

against distance. Empirical observations show that temperature range within distance is

largest in the Mahalanobis metric (Figure 3.2). Further, both Euclidean (Figure 3.2(a))

and Mahalanobis (Figure 3.2(c)) exhibit non-monotonic temperature rise. Therefore, we

choose the Chebyshev distance metric to determine a voxel’s distance from the exposed

needle. As shown in Figure 3.2(b), there is an exponential decrease in the temperature

with distance of the nature e−bd, where d is the distance of the voxel from the needle

position and b is a constant value.

Heat deposited dissipates with distance but increases with time until thermal equi-

librium is achieved. The explicit finite difference scheme computes the temperature at

a voxel using temperature information from its non-diagonal neighbours. As a result,

voxels at the same distance from the needle have different temperatures (Figure 3.3(a)).

Further, as seen in Figure 3.3(a), the average temperature of a voxel at a given dis-


0 10 20 30 40 50 60310

315

320

325

330

335

340

345

350

355

Distance (mm)

Avera

ge T

em

pera

ture

(K

)

1 minute20 minutesT = 320K

(a) Euclidean distance metric

0 10 20 30 40 50 60310

315

320

325

330

335

340

345

350

355

Distance (mm)

Avera

ge T

em

pera

ture

(K

)


(b) Chebyshev distance metric

0 10 20 30 40 50 60310

315

320

325

330

335

340

345

350

355

Distance (mm)

Avera

ge T

em

pera

ture

(K

)


(c) Mahalanobis distance metric

Figure 3.2: Average temperature per distance at 1 minute and 20 minutes using Eu-clidean, Chebyshev and Mahalanobis distance metrics for 30 mm needle exposure.


0 5 10 15 20 25 30310

315

320

325

330

335

340

345

350

355

Distance (mm)

Te

mp

era

ture

(K

)

20 minutesT = 320K

(a) Distance v. average temperature

0 50 100 150 200310

315

320

325

330

335

340

345

350

355

Time (s)

Te

mp

era

ture

(K

)

T = 320K

d = 1 mm

d = 2 mm

d = 3 mm

(b) Time v. average temperature for distance atd=1 to 3 mm

Figure 3.3: Relationship of time, distance, and temperature for a 30 mm needle usingthe Chebyshev distance metric.

tance increases exponentially with time as p− qe−at, where t is the time in seconds and

p, q, and a are unitless shape parameters (Figure 3.3(b)).

Now we can express the temperature at a voxel as a function of distance and time of

type (p − qe−at)e−bd. In order to estimate the parameters p, q, a, and b, we formulate a

least squares problem as follows:

minimizep,q,a,b

∑i∈τ

∑j∈T

(f(dj, ti)− pωbj + qωbjγ

ai

)2(dt-approx)

subject to p, q, a, b ≥ 0

where f(dj, ti) is the mean temperature of voxel j at distance dj and time ti, ωj = e−dj ,

γi = e−ti and τ is the set of time steps. Since needle position and orientation are known,

we can easily pre-compute dj and f(dj, ti) from BHTE.

Finally, we perform thermal dose optimization by finding the minimum time t required


to deliver the threshold dose to all the tumor voxels using a given needle position:

minimizet

t (dt-TDO)

subject to(p− qe−at

)e−bdj ≥ Dj ∀j ∈ T

t ≥ 0

where Dj is the required dose at voxel j, dj is the Chebyshev distance of voxel j from

the needle, and p, q, a, and b are the coefficients found using dt-approx. This model is

a convex optimization problem as it minimizes an affine function over a convex set. It

can thus be solved via an interior point method by minimizing the following log-barrier

objective function [10]:

minimizet

t− 1

v

∑j∈T

log((p− qe−at

)e−bdj −Dj

)− log (t)

where v is the barrier parameter. This unconstrained model is solved as a series of sub-

problems, where the value of v is gradually increased and the solution from the previous

iteration is used as a starting point. The constraints in dt-TDO are convex, and its

sum, representing a sum of thermal doses deposited on a target voxel from multiple

needles, is also a convex function. The dt-TDO model only considers target voxels and

not healthy voxels since addition of healthy voxels results in non-convex constraints. It is

worth noting that dt-TDO is generalizable to multiple needle treatment plans, however,

thermal dose deposition as temperature using multiple needles is not additive, and the

resulting solution would be inaccurate.

3.2.2 Approximation by isodose line

The radius of the ablation lesion does not increase after a state of thermal equilibrium is

reached (Figure 3.4). However, the probability of cell kill increases as the cell is exposed


0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

40

Time (s)

Th

resh

old

te

mp

era

ture

is

od

ose

lin

e d

iam

ete

r (m

m)

X−directionY−directionEquilibrium

(a) Single needle equilibrium at 250 s

0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

40

Time (s)

Th

resh

old

te

mp

era

ture

is

od

ose

lin

e d

iam

ete

r (m

m)

x−directiony−directionEquilibrium

(b) Clustered needle equilibrium at 356 s

Figure 3.4: Isodose line equilibrium for different needles

to a fixed temperature for longer periods of time. If we can obtain the outermost layer

of the lesion where voxels have reached threshold temperature α, we can safely assume

that all the voxels inside this layer are ablated. Thus, if we can identify the outermost

layer with temperature ≥ α, we can determine the radius of the lesion and the exposure

time needed to obtain that radius.

With a fixed needle position and orientation, we presolve BHTE for needles oriented

according to NOO with conducting tips varying 10-50 mm. At each time step of BHTE,

the outermost voxels that have temperature α ≥ 320 K are determined using the grassfire

algorithm [8]. The outermost layer is referred to as the threshold temperature α isodose

line or simply the isodose line

The diameter for the α isodose line is recorded for every time step, thus establishing

a rate of increase in the isodose line. Although we can fit an exponential curve using

these points and thereby optimize for treatment time, we note that the BHTE reaches

a state of thermal equilibrium after which the ablation diameter does not increase in

any direction (Figure 3.4). We record the time step at which BHTE reaches thermal

equilibrium along with the length of the diameter in each direction at that time step for

every needle length. This diameter is the largest lesion formed for a given needle under

ideal conditions, i.e., in the absence of large blood vessels that may cause a heat sink


Algorithm 5 iso-approx

Require: Threshold temperature α and time steps τ1: T ← Compute BHTE2: for t ∈ τ do3: P ← T (t, x, y, z) ≥ α4: Find P ′, the coordinates of outermost voxels of P , using the grassfire [8]5: dxt = max

(x,y,z)∈P ′x − min

(x,y,z)∈P ′x

6: dyt = max(x,y,z)∈P ′

y − min(x,y,z)∈P ′

y

7: dzt = max(x,y,z)∈P ′

z − min(x,y,z)∈P ′

z

8: if all the target voxels are covered then9: d = [dxt, dyt, dzt]

10: return (d/2, t)11: end if12: end for13: dx = max

t∈τdxt

14: dy = maxt∈τdyt

15: dz = maxt∈τdzt

16: t = arg mint∈τ

t | dxt = dx, dyt = dy, dzt = dz

17: d = [dx, dy, dz]18: return (d/2, t)

effect, and therefore represents a lower bound on the treatment time. The process to

obtain this lower bound and diameter is formalized in Algorithm 5, which assumes the

needle is parallel to the x-axis; the domain is rotated a priori to achieve this orientation

if necessary.

Algorithm 5 can be pre-computed for all the ablation needles, and thus the lower

bound on treatment time t and the ablation radii are known a priori for all needles.

The known ablation radii is used to determine the needle type necessary to treat the

entire tumor after which NOO is performed to obtain center and orientation. Although

ablation zone information is given by the needle manufacturer, the provided dimensions

are typically obtained from experiments performed on a bovine liver [17]. Thermal and

electrical properties of human liver and other organs can vary, and our isodose line

method provides physicians with a more accurate ablation zone while allowing for tissue


heterogeneity.

3.3 TDO with linear BHTE approximation

Our non-linear approximation oversimplifies the thermal distributions, matches the lesion

shapes to distance metric, and does not easily adapt to multiple needle ablation due to

the non-linearity as well as non-additive nature of temperature. Arrhenius damage index

is additive across multiple needles and also provides better damage information since

it uses thermal history of the voxel. As seen in Equation ATDM, the thermal damage

models are non-linear in nature. Therefore, we propose a linear relaxation of the model

where the coefficients are obtained a priori from Equation ATDM.

We define ui = [ci θi], a six-dimensional vector describing the ith needle position in

3D (ci) and orientation in 3D (θi) obtained from NOO-MVCE, NOO-MVCS, or NOO-

Kmeans. Let hjs be the thermal dose to voxel j in structure s. Given a fixed set of n

needles and their positions and orientations, the mathematical model to optimize thermal

dose received by every structure s is represented by the following linear TDO model where

the decision vector t = [t1, . . . , tn] is the amount of time (in seconds) for which needle i

deposits its thermal dose:

minimizet

∑s∈S

vs∑j=1

Fs(hjs) (Relaxed-TDO)

subject to hjs =n∑i=1

Djsuiti ∀s ∈ S ∪ T ,∀j = 1, . . . , vs

tmax ≤ ti ≤ tmin ∀i = 1, . . . , n

Fs(hjs) is the penalty incurred for dose hjs received by voxel j in structure s; vs is the

total number of voxels in structure s; Djsuiis the mean thermal dose deposited by needle

i with position and orientation given by ui in voxel j in structure s; and tmax and tmin

are the upper and lower limits on treatment time, respectively.


0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Arrhenius dose (hjs

)

Fs(h

js)

Healthy structureTarget structure

(a) Quadratic penalty, ws = 0.05 and ws = 0.95

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

Arrhenius dose (hjs

)

Fs(h

js)

α1 = 1

α1 = 0.5

α1 = 0.2

α2 = 3

α2 = 1

α2 = 0.5

(b) Exponential penalty for different α values

Figure 3.5: Quadratic (Equation QP) and exponential (Equation EP) penalties for targetsand OARs

The penalty in the Relaxed-TDO model can be formulated as a quadratic or an

exponential function. The quadratic penalty (QP, Figure 3.5(a)) is

Fs (hjs) =

ws(hjs −Hs

)2if hjs ≥ Hs ∀s ∈ H

ws (Hs − hjs)2 if hjs ≤ Hs ∀s ∈ T

0 if Hs ≤ hjs ≤ Hs ∀s ∈ H ∪ T

(QP)

where ws and ws are weights for under-dosing and over-dosing a voxel in structure s,

respectively, andHs andHs are the acceptable maximum and minimum dose for structure

s, respectively.

The exponential penalty (EP, Figure 3.5(b)) is

Fs (hjs) =

exp (α1 (Hs − hjs)) ∀s ∈ T

exp(α2

(hjs −Hs

))∀s ∈ H

(EP)

where α1 and α2 are weights for under- and over-dosing, respectively. The least hot

target voxel and the most hot OAR voxel will be penalized the most by the exponential

penalty. A similar exponential penalty function was used by Altrogge et al. [2] for target


voxels only.

The primary difference between the quadratic and exponential penalties lies in the

desired thermal spread within a structure. The quadratic penalty will approach its

minimum by trying to reach a thermal dose between upper (Hs) and lower (Hs) bounds

while the exponential penalty will seek a uniform thermal dose distribution. Both penalty

functions are convex and hence present an opportunity to use gradient-based convex

optimization algorithms. The Relaxed-TDO model has only box constraints, and is

solved using a projected gradient algorithm with Goldstein-Armijo line search [25].

3.4 Voltage-based TDO

BHTE non-linear approximations do not yield very promising results since solutions ad-

here to shape of the distance metric and it does not scale to multiple needle ablations.

While isodose approximation provides promising results, it requires a constant needle

temperature, which is not realistic as needle temperature increases steadily through out

the treatment. Further, although linear approximation is intuitive, it does not represent

the true physical process of thermal deposition. Additionally, due to localized dose de-

position, the length of treatment does not affect its quality, and we therefore propose a

simpler approach to understanding thermal damage and treatment times using an ex-

haustive search across needle configurations for different damage models. We introduce a

voltage-based thermal dose computation where we compute thermal doses with increasing

voltages and hence temperatures (Algorithm 6).

We define a set of needle configurations as a combination of needle type (n ∈ N )

and source voltage (φ ∈ V). The set of damage models is given by d ∈ D = BHTE,

ATDM, D63, D70, D80, D95. For each needle configuration (φ, n), we first compute

the BHTE for a fixed treatment time using inputs from Laplacian and then compute the

ATDM followed by the percentage damage models if required for thermal damage d. We


Algorithm 6 Focal ablation framework

Require: N ← Set of needles typeRequire: V ← Set of source voltagesRequire: d ∈ D ← Damage modelRequire: S ← Set of structures

1: for n ∈ N do2: Sn ← Perform NOO3: for φ ∈ V do4: L ← computeLaplacian(Sn, φ) . Solve Laplacian5: HBHTE(φ, n, t) ← computeBHTE(L) . Solve BHTE6: if d ∈ D = ATDM, D63, D70, D80, D95 then7: HATDM(φ, n, t) ← computeArrhenius(HBHTE(φ, n, t)) . Solve ATDM8: if d ∈ D = D63, D70, D80, D95 then9: Hd(φ, n, t) ← computePercentDamage(HATDM(φ, n, t))

10: end if11: end if12: td∗(φ, n)← solve Minimum treatment time13: end for14: φd∗n ← solve Minimum voltage15: end for16: (φd, nd)∗ ← Get best needle configuration (Algorithm 7)17: return

[(φd, nd)∗, td∗(φd, nd)

]save this information to determine the minimum treatment voltage and treatment times.

We define a numerical dose structure Hd(φ, n,xj, t) to identify damage using model d to

voxel xj ∈ H ∪ T at time t due to needle configuration (φ, n).

For a fixed treatment time tmax, the minimum treatment voltage for full target cov-

erage using damage model d ∈ D and needle type n ∈ N is given by

φd∗n = minimizeφ

φ (Minimum voltage)

subject to Hd(φ, n,xj, tmax) ≥ HdTH ∀j ∈ T , φ ∈ V

where HdTH is the threshold damage value for model d, and Hd(φ, n,xj, tmax) is the damage

to voxel xj at tmax minutes when using needle configuration (φ, n). If temperature is used

to quantify thermal damage, then HdTH = TTH = 60C; if the Arrhenius damage index

is used to quantify thermal damage, then HdTH = ΩTH = 1; and if p percent damage is


Algorithm 7 Best needle configuration

Require: N ← Set of needles typesRequire: V ← Set of source voltagesRequire: d ∈ D ← Damage modelRequire: Hd

TH ← Threshold dose valueRequire: Hd(φ, n,xj, tmax)← dose structure

1: for n ∈ N do2: for φ ∈ V do3: T = xj | xj ∈ T and Hd(φ, n,xj, tmax) ≥ Hd

TH . Target damage

4: H = xj | xj ∈ H and Hd(φ, n,xj, tmax) ≥ HdTH . OAR damage

5: ptargetφ,n ← 100× |T |/|T | . Percent target damage

6: pOARφ,n ← 100× |H|/|H| . Percent OAR damage

7: end for8: end for9: (φ, n)∗ ← argminpOAR

φ,n : ptargetφ,n ≥ 99.99%

10: return (φ, n)∗

used to quantify thermal damage, then HdTH = p%.

Similarly, for a fixed voltage φ, the minimum treatment time for full target coverage

using damage model d ∈ D and needle type n ∈ N is given by

td∗φ,n = minimizet

t (Minimum treatment time)

subject to Hd(φ, n,xj, t) ≥ HdTH ∀j ∈ T

t ≥ 0

To choose a single best needle configuration for damage model d, we select the needle

configuration with 100% coverage and the least OAR damage (Algorithm 7).

Chapter 4

Needle deflection analysis

A needle may unpredictably deflect on its path, potentially impacting target and OAR

damage. To understand this impact, we perform scenario-based needle deflection analysis

on target and OAR damage. We use NOO-MVCE and NOO-Kmeans from Chapter 2

to identify needle positions and orientations for single and multiple needle placement,

respectively, referred to as the base case scenario. Around these base case needle centers,

we identify potentially deflected centers within a 2-4 mm radius [22]. We create scenarios

for (1) translational deflection, where only the center is deflected from the base case, and

(2) combined deflection, where both the center and orientation are deflected.

Let ci and θi be the base case center and orientation for needle i, respectively. We

identify a set of voxels, Lci , that are between [rmin, rmax] from the base case center ci

(Figure 4.1(a)). For k ablations, setM = Lc1×· · ·×Lck represents all the combinations

of the translational needle deflection scenarios. Translational deflection for a single needle

ablation (k = 1) has |M| = |Lc1 | scenarios, while for multiple (k) needle ablation, the

number of scenarios is |M| = |Lc1 | × · · · × |Lck |. For our test cases, there are 33-35,937

translational scenarios.

For combined deflection analysis, we define a set of deflection angles Γ where γ ∈ Γ

represents the angle with the base case scenario θ. To obtain these deflected orientations,

45

Chapter 4. Needle deflection analysis 46

34

35

36

37

33

34

35

3646

47

48

49

xy

z

Base case centerDeflected centers

(a) Sample deflected centers

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

Base case orientationDeflection orientations

(b) Sample deflected orientations

Figure 4.1: Deflection scenarios

we assume that the needle must pass through one of the target voxels. Position vectors

of these target voxels from the geometric centroid define the new deflected orientation if

its angle with base case orientation, θ, is given in deflection angle set Γ (Figure 4.1(b)).

Thus, for each needle i, there are O = Γ × Lci scenarios, resulting in |O1| × . . . × |Ok|

combination scenarios. For our test cases, there are 132-3,000,000 scenarios for |Γ| = 4.

As small targets typically require single needle ablation, the overall problem size per

scenario is tractable and we can enumerate all deflection scenarios. However, a medium-

to-large target requires multiple needle ablation, and thus the number of scenarios is

exponentially larger, which is compounded by more computationally intensive scenar-

ios. For these cases, we sample scenarios using a Gaussian distribution (Figure 4.2) to

approximate the likelihood of a scenario.

Algorithm 8 shows the steps to approximate the likelihood of needle i deflecting to vi

from ci. First, we determine the Euclidean distance of voxel vi from the base case center

ci. Next, the position of voxel vi with respect to the base case center ci is determined

by the angle ω1 between its position vector and the base case orientation θi:

y(vi) = sign(α)×√

(v1 − c1)2 + (v3 − c3)2 + (v3 − c3)2

rmax

∀vi ∈ Lci


−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized deflection distance

Lik

elihood o

f occurr

ence

Figure 4.2: Likelihood of a scenario occurrence

P (y(vi)) ∈ (0, 1) is normally distributed with mean y and standard deviation σ, respec-

tively, for y ∈ (−1, 1), which is a vector of size |Lci| whose each entry corresponds to a

normalized Euclidean distance of voxel vi from base case center ci. Finally, for a scenario

s = (vi, . . . ,vk), we approximate its likelihood of occurrence P (s) by

P (s) =P (s)

max∀s∈M

P (s)

where

P (s) =k∑i=1

P (vi)

If deflection scenarios cannot be enumerated, we sample all scenarios with P (s) > 0.5.


Algorithm 8 Define deflected centers and their likelihood of occurrence

Require: V← coordinates for all target voxelsRequire: θ ← orientation of the needleRequire: c← coordinates of needle centerRequire: k ← number of ablationsRequire: rmin, rmax ← minimum and maximum deflection radius, respectively

1: Lc ← ∅2: Y ← ∅3: for v ∈ V do4: d←

√(v1 − c1)2 + (v3 − c3)2 + (v3 − c3)2

5: if d ≥ rmin and d ≤ rmax then6: u← (v − c)/||v − c||227: α← cos−1 (θ.u/(||θ||.||u||))8: y ← sign(α)× (d/rmax)9: Y ← Y ∪ y

10: Lc ← Lc ∪ v11: end if12: end for13: if k < 2 then14: p← 1/|Lc| . a vector of P(v ∈ Lc)15: else16: µ← mean(Y )17: σ ← std(Y )18: p← N (µ, σ)19: end if20: return Lc,p

0 0.5 1 1.5 2 2.5 32

2.5

3

3.5

Base case center is at [2, 2], deflected voxel is [3, 3.5]

ω1

Figure 4.3: Position vector angle

Chapter 5

Results and discussion

We discuss the results for our convex NOO models, TDO model that uses dt-approx and

iso-approx based BHTE approximation, followed by discussion on these results on 2D

slices (Section 5.1). Results for our NOO models with and without trajectory planning,

on 3D liver clinical cases followed by the TDO results for these needle placements appear

in Section 5.2. Finally, we discuss the impact of translational and combined needle

deflection on target and OAR damage in Section 5.3.

Our computations are performed on MATLAB R2008b or MATLAB R2015b for both

NOO and TDO. We use CVX, a package for specifying and solving disciplined convex

optimization problems [18, 28], to solve the NOO-MVCE and NOO-MVCS, and Gurobi

[29] to solve RFA-SCP and RMP-B with FC-B.

5.1 2D results

We test our approach on 2D slices of randomly generated cases (Table 5.1). The tumor

slices are generated so that different needle sizes are required for treatment. Cases 1-4

are small target slices where a single needle with an ellipsoid ablation zone is sufficient

for treatment, and Cases 5-8 are medium slices where a cluster of three needles with

a spherical ablation zone is required to ablate all the target voxels. The domain sizes

49

Chapter 5. Results and discussion 50

Table 5.1: Generated 2D case studies

IDSize Target Boundary OAR Target area

(mm × mm) voxels target voxels voxels (mm2)

1 23 × 12 205 50 14195 2052 30 × 10 228 63 14172 2283 39 × 12 404 84 13996 4044 49 × 14 579 102 13821 579

5 30 × 29 694 84 13706 6946 32 × 31 830 91 13570 8307 32 × 29 776 88 13624 7768 40 × 29 929 100 13471 929

for every case are fixed to 120 mm × 120 mm with 1 mm × 1 mm voxel size. We

can empirically estimate the needle length a priori as we have presolved BHTE and

know the maximum lesion length in each direction. A voxel is considered ablated if it

receives thermal dose with temperature ≥ 320 K and the damage to non-target voxels is

considered as damage to OAR voxels.

5.1.1 NOO

Models NOO-MVCE and NOO-MVCS are solved under two scenarios: (1) all target

voxels are considered, and (2) only boundary target voxels, obtained using grassfire [8],

are considered. Table 5.2 shows the runtimes for these scenarios for all the cases as well

as the computational improvement from only considering boundary target voxels instead

of all target voxels. NOO-MVCE is solved to optimality in <4 s using all target voxels,

and <1 s using only boundary target voxels, resulting in 62% average computational

gain. For NOO-MVCS, nearly all scenarios are solved to optimality in <0.5 s, with only

using boundary target voxels yielding 45% average computational gain. Table 5.3 shows

the needle results for NOO-MVCE and NOO-MVCS. These numbers are the same when

NOO is solved using all target voxels or only boundary target voxels. Figure 5.1 shows

the needle position in the target.


Table 5.2: Computation time results for NOO-MVCE and NOO-MVCS

IDAll target Boundary Boundary target

NOO modelvoxels (s) target voxels (s) voxel improvement

1 1.72 0.78 55%

NOO-MVCE2 1.87 0.87 54%3 3.21 1.02 68%4 3.48 1.00 71%

5 1.80 0.31 83%

NOO-MVCS6 0.47 0.34 29%7 0.49 0.34 31%8 0.47 0.31 35%

Table 5.3: Needle results for NOO-MVCE and NOO-MVCS. θ: orientation of the needle,r: radius of the fitted ellipse or sphere.

ID c Ablated area (mm2) λ (mm) θ r (mm) NOO model

1 (60,60) 218 20 (1,0) (6.00,11.58)

NOO-MVCE2 (60,60) 240 30 (1,0) (5.00,15.31)3 (60,60) 424 40 (1,0) (6.58,20.48)4 (60,60) 589 50 (1,0) (7.50,25.00)

5 (60,60) 707 20 (1,0) 15.00

NOO-MVCS6 (60,60) 883 30 (1,0) 16.767 (60,60) 883 30 (1,0) 16.768 (60,60) 1257 30 (1,0) 20.00


(a) Case 1, 205 mm2 (b) Case 5, 707 mm2

(c) Case 2, 228 mm2 (d) Case 6, 883 mm2

(e) Case 3, 404 mm2 (f) Case 7, 883 mm2

(g) Case 4, 579 mm2 (h) Case 8, 1257 mm2

Figure 5.1: Slices illustrating NOO-MVCE, NOO-MVCS, and dt-approx, iso-approx andactual full BHTE calculations. The non-gray region inside an iso-dose line indicates itsOAR coverage.


Table 5.4: Optimal parameter values for dt-approx for different needle lengths

Single needle Clustered needles

Parameter 20 mm 30 mm 40 mm 50 mm 20 mm 30 mm

p 30.92 29.46 28.62 27.89 56.25 57.30q 24.35 23.32 22.77 22.26 45.23 46.63a 0.02 0.02 0.02 0.02 0.02 0.02b 0.32 0.3 0.29 0.27 0.29 0.28

R2 0.99 0.99 0.99 0.99 0.99 0.99RMSE 0.27 0.28 0.29 0.31 0.79 0.91SSE 3939.31 4181.6 4413.7 5026.9 37579 49939

5.1.2 TDO

We use MATLAB’s fmincon solver to solve the dt-approx model because it is a non-

convex model. We use the interior-point solver of fmincon instead of its sequential

quadratic (SQP) solver as the interior-point solver takes advantage of more accurate

user-defined derivative and Hessian information than the SQP solver, which uses a quasi-

Newton approximation of the Lagrangian. Once we obtain parameters (Table 5.4), we

solve dt-TDO using an interior point method. Figure 5.2 shows the original and the

fitted curves.

To assess the quality of BHTE approximations, we use three conformity indices: (1)

Dice similarity coefficient (DSC) [19], (2) classic conformity index (CI) [34], and (3)

Paddick’s conformity index (PI) [46]. Let Va and Vb be the set of voxels covered by the

approximation and BHTE, respectively. These indices are then defined as follows:

DSC =2|Va ∩ Vb||Va|+ |Vb|

(DSC)

CI =|Va||Vb|

(CI)


BHTE (original curve) dt-approx (fitted curve)

SingleNeedle

ClusteredNeedles

Figure 5.2: Heat maps for a 30 mm needle in a 120 mm × 120 mm domain where voxelsize is 1 mm × 1 mm and the time step is 0.1 s


Table 5.5: Conformity of BHTE approximations to actual BHTE. DSC = Dice similaritycoefficient, CI = classic index, PI = Paddick index

IDdt-approx iso-approx

Runtime (s) DSC CI PI Runtime (s) DSC CI PI1 3.89 0.62 1.22 0.39 0.06 0.95 0.90 0.902 2.56 0.56 1.09 0.32 0.06 0.93 0.86 0.863 2.93 0.57 1.14 0.32 0.07 0.94 0.88 0.884 2.63 0.56 1.13 0.31 0.08 0.93 0.87 0.875 4.60 0.88 1.13 0.77 0.09 0.93 0.87 0.876 3.54 0.92 1.11 0.86 0.02 0.89 0.80 0.807 3.47 0.87 1.30 0.77 0.01 0.88 0.79 0.798 3.47 0.87 1.30 0.77 0.08 0.91 0.83 0.83

PI =|Va ∩ Vb|2

|Va| × |Vb|(PI)

Both DSC and PI measure the similarity between two sets and are able to compute actual

overlap between them. PI measures the overlap by multiplying the conformity with

respect to approximation and BHTE. CI does not consider the actual overlap between

approximation and BHTE and hence may have larger values if the BHTE covers fewer

voxels than the approximation.

Table 5.5 contains the conformity indices and computation times of dt-approx and iso-

approx, while Figure 5.1 shows the isodose lines including the isodose line from actual

BHTE from treatment time specified by TDO using the iso-approx method. The dt-

approx is unable to capture the thermal lesion shape when using single or multiple needle

and conforms to the shape of the distance metric (Figure 5.3), resulting in poor conformity

indices to the actual BHTE lesion. The iso-approx algorithm is able to capture the BHTE

lesion shape as well as the state of equilibrium resulting in clinically acceptable conformity

indices.

In order to measure the treatment quality, we compute conformity of dt-approx,

iso-approx, and BHTE with the target itself using the conformity indices described pre-

viously, where Va is now the set of target voxels (Table 5.6). We also compute the per-


10

10

10

10

20

20

20

20

20

20

30

30

30 30

30

30

3030

40

40

40 40

40

40

40

50

50

50

50

50

50

20 40 60 80 100 120

10

20

30

40

50

60

70

80

90

100

110

120

(a) Single needle

10

10

10

10

20

20 20

20

2020

30

30

30 30

30

30

3030

40

40

40 40 40

40

40

404040

50

50

50

20 40 60 80 100 120

10

20

30

40

50

60

70

80

90

100

110

120

(b) Multiple needles

Figure 5.3: Chebyshev distance maps (mm)

centage of tumor and OAR cells covered by the dt-approx, iso-approx, and BHTE. The

treatment times in Table 5.6 under the dt-approx and iso-approx columns are rounded

up to the closest minute.


Tab

le5.

6:T

reat

men

tqual

ity

met

rics

show

ing

per

centa

geof

targ

etan

dO

AR

voxel

sre

ceiv

ingα≥

320

Kan

dco

nfo

rmit

yof

the

trea

tmen

tpla

nw

ith

resp

ect

toth

eta

rget

.B

HT

Eis

obta

ined

usi

ng

iso-

appro

xtr

eatm

ent

tim

e.

IDdt-

appro

xis

o-ap

pro

xB

HT

ET

reat

men

tO

AR

Tar

get

DSC

CI

PI

Tre

atm

ent

OA

RT

arge

tD

SC

CI

PI

OA

RT

arge

tD

SC

CI

PI

tim

e(m

in)

(%)

(%)

tim

e(m

in)

(%)

(%)

(%)

(%)

12.

01.

2673

.17

0.56

0.62

0.33

2.0

0.27

100

0.92

0.84

0.84

0.46

100

0.86

0.76

0.76

21.

01.

2863

.16

0.52

0.70

0.28

2.0

0.33

100

0.91

0.83

0.83

0.64

100

0.83

0.71

0.71

32.

02.

4366

.58

0.53

0.66

0.29

5.0

0.79

100

0.88

0.78

0.78

1.28

100

0.82

0.69

0.69

42.

03.

0257

.34

0.50

0.77

0.25

5.0

0.56

94.6

50.

910.

930.

831.

1497

.41

0.87

0.80

0.76

51.

01.

0791

.64

0.86

0.89

0.74

6.0

0.78

100

0.93

0.87

0.87

1.69

100

0.86

0.75

0.75

62.

03.

2699

.88

0.79

0.65

0.65

2.0

0.52

100

0.96

0.92

0.92

2.14

100

0.85

0.74

0.74

71.

03.

6310

00.

760.

610.

612.

00.

4610

00.

960.

920.

932.

0810

00.

850.

730.

738

1.0

2.54

100

0.84

0.73

0.73

3.0

0.99

100

0.93

0.87

0.87

2.57

100

0.84

0.73

0.73


5.1.3 Discussion

Using the empirical data from BHTE and vendor specifications, we are able to choose the

needle type for the treatment based on percentage of OAR and target structure coverage.

The ratio of OAR-to-target coverage can help us choose an appropriate needle type (e.g.,

single or clustered). Clinically, this decision will implicitly prevent unnecessary damage

to OARs.

We solve BHTE using the input from NOO and approximate it by two methods. Using

mean temperatures for dt-approx method causes the temperatures of voxels to be the

same at a given distance because using a finite difference scheme for BHTE gives a varied

temperature profile for such voxels, resulting in conformity of the approximated lesion to

the distance metric used. We verify this phenomenon empirically and as seen in Figure

5.1, the dotted rectangular region conforms to a Chebyshev distance metric (Figure

5.3). The shorter treatment time for the dt-approx method is due to the exponential

approximation that rapidly increases temperature with time which in turn fails to capture

thermal equilibrium, resulting in poor conformity.

The iso-approx algorithm tracks the radius of the isoline at each time step and hence

is able to capture the thermal equilibrium giving better conformity indices. Further,

the algorithm terminates when maximum lesion size is reached or if all the target voxels

are covered with shorter treatment times. The iso-approx algorithm is a geometrical

approximation to actual BHTE and will usually provide an underestimation, resulting in

an underestimation of OAR coverage compared to BHTE and dt-approx.

We note that our approach yields short treatment times relative to clinical practice.

Heat spreads in the tissue until thermal equilibrium is reached, and high temperatures

cause rapid tissue desiccation, typically in minutes [1]. Despite this rapid temperature

rise, clinical treatment times are much longer to ensure tumor kill. Further, since we

have ignored the power absorption term, the temperature rise is somewhat faster than

when the power absorption term is included. This higher temperature rise is due to


the constant high temperature, starting with t = 0, of the voxels in contact with the

needle, whereas with the power absorption term, all the voxels are initialized to body

temperature causing a steady temperature rise. In this work, we have rounded up the

seconds to the closest minute as the ablation devices are typically designed to use minutes

as the smallest unit.

Despite some interesting results, the non-linear approximation to BHTE is a gross

oversimplification of actual thermal processes beginning with an assumption of constant

temperature, which is unrealistic as needle temperature, and therefore heat deposition,

steadily increases with time until a point of thermal equilibrium. Further, the models

are difficult to adapt to multiple needle ablation due to non-linearity as well as lack of

clear geometric shapes for randomly placed multiple needles.

5.2 3D results

We perform experiments on liver cases (Table 5.7) obtained from Robarts Research In-

stitute, Western University. In a clinical setting, tumors are over-ablated to ensure

microscopic tumor particles are killed along with the target itself. Therefore, we add

surgical margins of 0 mm (N), 3 mm (S), 5 mm (M), and 10 mm (L) around the target.

Further, in liver ablation OAR sparing is insignificant due to its regenerative properties,

and thus no explicit OAR margin is added to the target. However, we consider damage

to non-target voxels outside surgical margin as OAR damage to understand the impact

of input parameters. Finally, these liver targets were originally treated using MWA and

therefore, we are unable to compare the treatment quality of our framework with clinical

results.


Table 5.7: Description of case studies

IDVolume Target Boundary(mm3) voxels target voxels

100% 50% 100% 50% 100% 50%

1N 898 544 898 544 501 2931S 3063 1641 3063 1641 840 4301M 5138 2595 5138 2595 1106 8081L 13003 5874 13003 5874 1911 1839

2N 4657 2090 4657 2090 1591 7382S 10595 4430 10595 4430 2178 7472M 15595 6320 15595 6320 2610 10582L 32225 12327 32225 12327 3830 2084

3N 13481 5183 13481 5183 3273 13343S 24895 9169 24895 9169 4060 18163M 33881 12226 33881 12226 4640 21583L 61771 21442 61772 21442 6230 3070

Table 5.8: Needle types (N )

Abbreviation Needle description Active tip length (mm)

SN7 Single needle 7SN10 Single needle 10SN20 Single needle 20SN30 Single needle 30CN25 Clustered needle 25MN2K30 Multiple needle, k = 2 30MN3K30 Multiple needle, k = 3 30MN3K40 Multiple needle, k = 3 40

5.2.1 NOO: MVCE, MVCS, and NOO-Kmeans

Similar to 2D cases, we test our NOO approach for all target voxels and for only boundary

target voxels. Additionally, for faster computation, both scenarios are solved for the

entire target as well as targets sampled at 50%. We restrict our needle types, N , to eight

Covidien specifications (Table 5.8) [38].

As the number of voxels increases, the runtimes for both NOO-MVCE and NOO-

MVCS increase (Figure 5.4). For unsampled cases, the model NOO-MVCS runs in <8

s or under 8 s and <3 s for all target and boundary target voxels, respectively. Using


Table 5.9: Numerical results for NOO. A(ξ) = Fitted volumes, c = needle center, θ =needle orientation.

ID A(ξ) (mm3) c θ Model

1N 956[35.24 34.39 47.15

]> [−0.87 −0.5 0.03

]>NOO-MVCE

2N

[3922.63929.6

] [41.23 36.67 40.6843.35 46.95 40.19

]> [−0.1 0.1 0.030.82 −0.57 −0.01

]>NOO-Kmeans

3N 48419[47.79 48.17 69.59

]> [1 0 0

]>NOO-MVCS

3N

9881.49536.49519.6

42.62 54.58 70.3955.87 50.00 70.1445.65 39.90 70.45

> 0.8 0.59 −0.060.16 −0.99 0.03−0.94 0.39 0.02

> NOO-Kmeans

boundary target voxels gives an average computational gain of 60% and 53% for NOO-

MVCS for unsampled and sampled cases, respectively. For all unsampled target voxels,

NOO-MVCE does not finish in reasonable amount of time (>1 hour) for Cases 2L and 3M,

while Case 3L runs out of memory (Figure 5.4(c)). However, NOO-MVCE runs in under

a minute for unsampled boundary target voxels in all cases. An average computational

gain of 81% and 83% is obtained for unsampled and sampled cases, respectively, when

only boundary voxels are used for NOO-MVCE.

Given the computational advantage of using boundary voxels, we use boundary voxels

to solve MVCE for NOO-Kmeans. For unsampled cases, NOO-Kmeans runs in < 1 min

(Figure 5.4(e)), while sampled cases are all <10s (Figure 5.4(f)). These fast runtimes

may appear counterintuitive since NOO-MVCE is solved k times, once for each cluster.

However, each cluster contains only a subset of target voxels, and we consider only the

boundary voxels of these clusters.

When using boundary voxels, runtimes are under a minute for the largest unsampled

case (Case 3L) for all NOO methods. Therefore, we report results only for unsampled

cases for both NOO and TDO. For selected cases, Figure 5.5 shows the needle orientations

given by NOO-MVCE, NOO-MVCS, and NOO-Kmeans models, and Table 5.9 shows

their fitted volumes.


N S M L0

1

2

3

4

5

6

7

8

Surgical margin

Ru

ntim

e (

s)

Case 1 (all target voxels)

Case 1 (boundary target voxels)





(a) MVCS (unsampled)

N S M L0

1

2

3

4

5

6

7

8

Surgical margin

Ru

ntim

e (

s)







(b) MVCS (50% sampled)

S M L

100

200

300

400

500

600

700

800

Surgical margin

Runtim

e (

s)







(c) MVCE (unsampled)

N S M L0

100

200

300

400

500

600

700

800

Surgical margin

Runtim

e (

s)







(d) MVCE (50% sampled)

N S M L0

10

20

30

40

50

60

Surgical margin

Ru

ntim

e (

s)

Case 1, k = 2

Case 1, k = 3

Case 2, k = 2

Case 2, k = 3

Case 3, k = 2

Case 3, k = 3

(e) NOO-Kmeans (unsampled, boundary voxels)

N S M L0

10

20

30

40

50

60

Surgical margin

Runtim

e (

s)

Case 1, k = 2

Case 1, k = 3

Case 2, k = 2Case 2, k = 3

Case 3, k = 2

Case 3, k = 3

(f) NOO-Kmeans (50% sampled, boundary vox-els)

Figure 5.4: Runtimes for NOO models NOO-MVCE and NOO-MVCS


(a) Case 1N single needle (NOO-MVCE) (b) Case 3N single clustered needle (NOO-MVCS). The dotted lines are the equilat-eral triangle whose vertices correspond tothe centers of the conducting tines in thecluster.

(c) Case 2N two needles (NOO-Kmeans) (d) Case 3N three needles (NOO-Kmeans)

Figure 5.5: Needle positions and orientations


Table 5.10: Total ellipses generated

Case IDTip length (λ mm)

7 10 20 30 40

1N 396 169 - - -1S 831 437 37 - -1M 1182 775 72 - -1L 1572 1246 228 4 -

2N 1599 986 254 7 -2S 2211 1478 440 61 -2M 2606 2037 783 128 -2L 2696 2270 1147 339 33

3N 2780 1927 979 356 223S 3649 2735 1236 442 763M 3740 3001 1577 642 1683L 4430 3712 2271 953 291

5.2.2 NOO: Trajectory planning

We consider five needle tip lengths (mm), Λ = 7, 10, 20, 30, 40. For each of the

12 cases, we generate ellipses for each tip length, resulting in a maximum of 60 runs

(12 cases × 5 needle tips). We control the number of ellipses generated with an upper

bound. Therefore, while the maximum number of ellipses generated is |C| × |Θ|, the

actual number of ellipses generated is less due to elimination of invalid ellipses (Table

5.10). We note that certain combinations, e.g., Case 1N and λ ≥ 20 mm or Case 2N

and λ ≥ 40 mm, do not produce any valid ellipses because the needle tip length is longer

than the target size resulting in intersection with the OAR voxels. Thus, although in

general increasing numbers of ellipses increases runtime, the actual runtime is affected

by preprocessing to reject invalid ellipses and paths (Figure 5.6).

Figure 5.7 illustrates the computational complexity of generating invalid ellipse pairs

using pairwise no-good and group cuts. The maximum number of pairwise no-good cuts

and group cuts that can be generated is m!/(2× (m− 2)!) and m, respectively. For our

cases, the number of no-good cuts is up to 3,000,000 and therefore, for large values of m,


1000 2000 3000 4000 5000 6000 70000

50

100

150

200

250

300

350

Maximum ellipses

Ru

ntim

e (

min

)

Figure 5.6: Ellipse create runtime (min)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5

3x 10

6

Number of ellipses

To

tal cu

ts g

en

era

ted

Pairwise cutsGroup cuts

(a) Total cuts

0 1000 2000 3000 4000 50000

50

100

150

Number of ellipses

Ru

ntim

e (

min

)

Pairwise cutsGroup cuts

(b) Cut creation runtime

Figure 5.7: Comparison of pairwise validity cuts

creating all invalid ellipse pairs using no-good cuts is intractable due to memory and time

constraints. However, creating invalid ellipse pairs using group cuts is computationally

tractable with up to only 4,500 cuts created for our cases. Therefore, we present results

using only group cuts.

Total ablations and model feasibility depend primarily on the definition of ellipse sets

(Table 5.11). When k = ∞, the solution provides a lower bound (`) on the minimum

ablations required for full target coverage. When k ablations are desired, the models may

be infeasible due to no pairwise valid needle positions or incomplete target coverage. For

instance, for Case 1N, ` = 1 ablations are required for λ ≥ 7 and for k = 2 > `, the


Table 5.11: Total ablations with k =∞ (RMP-B)

Case IDTip length λ mm

7 10 20 30 40

1N 1 1 - - -1S inf 2 1 - -1M inf 3 1 - -1L inf inf 1 1 -

2N 4 3 1 1 -2S inf 4 1 1 -2M inf 7 1 1 -2L inf inf 2 1 1

3N inf 6 2 1 13S inf 9 2 1 13M mem 0 3 2 23L mem 0 4 2 2

model is infeasible due to intersecting needle placements. For Case 2S, ` = 4 ablations

are required for λ = 10 and hence when k ∈ 1, 2 < `, the model becomes infeasible due

to insufficient target coverage.

The row generation on the target coverage (RMP-E + FC-E) performs 67% times

faster then RFA-SCP on more than 60% of our cases (Figure 5.8(a)); while, as seen

in Figure 5.8(b), the model RFA-SCP outperforms row generation on pairwise validity

matrix (RMP-B + FC-B). The runtime of the model is influenced by the target size as

well as number of ablations desired. RMP-B + FC-B performs poorly because the RMP

is resolved at each iteration with new cuts when more than a single ablation is required.

However, RMP-E + FC-E outperforms RFA-SCP and RMP-B + FC-B because of up to

90% reduction in target coverage constraints (E), and in a multi-needle ablation scenario,

the possibility of invalid needle placements is higher than the failure to cover internal

target voxels due to a large number of needle combinations. Finally, although using

k = ∞ does not require a priori knowledge of the number of ablations, faster runtimes

(up to < 50 min) can be achieved by bounding k, especially to detect infeasible or

undesirable solutions (e.g., large number of ablations), with row generation on target


0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350

RFA−SCP Runtime (min)

RM

P−

E +

FC

−E

Ru

ntim

e (

min

)

(a) Target coverage

0 2000 4000 6000 8000 100000

1000

2000

3000

4000

5000

6000

7000

8000

9000

RFA−SCP Runtime (min)

RM

P−

B +

FC

−B

Runtim

e (

min

)

(b) Pairwise validity

Figure 5.8: Comparison between row generation approaches with RFA-SCP model

coverage outperforming the full model (Figure 5.9).

5.2.3 TDO

We solve BHTE and Laplacian using a finite difference scheme (Section 3.1) to obtain

thermal distributions over a 20 min simulation with a 0.5 times step for eight needle

types (Table 5.8) with source voltages, V , varying from 2.5 V to 30 V in increments of

2.5 V. Thermal distributions are computed only if non-intersecting needle positions are

found. Each of the 12 cases consists of 96 runs (8 needle types × 12 source voltages), for a

total of 1,152 runs. However, when performing TDO for needle placements obtained with

trajectory planning, only the needle types where model RMP-B was able to find a feasible

needle placement were explored. Hence, TDO with trajectory planning is performed for

fewer needle types than non-trajectory placement.

The computational runtime of each run is the total time to solve Laplacian, BHTE,

and ATDM; computational time is largely driven by the Laplacian (Figure 5.11(a)). We

assess target and OAR damage using the following thermal damage models: (1) ≥60C

threshold temperature from BHTE (T60), (2) ATDM, (3) 63% damage (D63), (4) 70%

damage (D70), (5) 80% damage (D80), and (6) 95% damage (D95).


0

50

100

150

200

250

300

Cases

Runtim

e (

min

)

1N

71

N1

01

S7

1S

10

1S

20

1M

71

M1

01

M2

01

L7

1L

10

1L

20

1L

30

2N

72

N1

02

N2

02

N3

02

S7

2S

10

2S

20

2S

30

2M

72

M1

02

M2

02

M3

02

L7

2L

10

2L

20

2L

30

2L

40

3N

73

N1

03

N2

03

N3

03

N4

03

S7

3S

10

3S

20

3S

30

3S

40

3M

10

3M

20

3M

30

3M

40

3L

30

3L

40

k =∞

k = 6k = 1k = 2k = 3

(a) RFA-SCP

0

50

100

150

200

250

300

350

Cases

Runtim

e (

min

)

1N

71N

10

1S

71S

10

1S

20

1M

71M

10

1M

20

1L7

1L10

1L20

1L30

2N

72N

10

2N

20

2N

30

2S

72S

10

2S

20

2S

30

2M

72M

10

2M

20

2M

30

2L7

2L10

2L20

2L30

2L40

3N

73N

10

3N

20

3N

30

3N

40

3S

73S

10

3S

20

3S

30

3S

40

3M

10

3M

20

3M

30

3M

40

3L30

3L40

k =∞

k = 6k = 1k = 2k = 3

(b) RMP-E + FC-E

Figure 5.9: Computational runtimes with bounded k


(a) Case 1N, λ = 7 mm (b) Case 2S, λ = 10 mm

Figure 5.10: Needle placement for single and multiple needle ablation using trajectoryplanning

The maximum temperature in the target increases with an increase in source voltage;

at least 7.5 V is recommended for the illustrated Case 1N (Figure 5.11(b)). High source

voltage increases the numerical value of the initial conditions for the Laplacian, caus-

ing high target temperatures, while longer or multiple needles increase the needle-voxel

intersection set, resulting in larger thermal spread. Hence, more needles or high source

voltage yield large ablation volumes (Figure 5.12) and high target (Figure 5.13) and OAR

damage (Figure 5.14), and consequently high tissue molecular damage.

Full coverage is seen when more needles operate at low voltage or fewer needles

operate at high voltage. Further, a low and high source voltage is recommended when

damage is quantified by BHTE and D95 models, respectively, resulting in a different

needle configurations for the same case. This difference in needle configuration arises

because tissue molecular damage increases with the duration of exposure to temperatures

≥60C, and BHTE damage occurs before D95 damage (Figure 5.15). Therefore, certain

needle configurations achieve full BHTE coverage but partial D95 coverage because all

the target voxels are not exposed long enough at temperatures ≥60C. Thus, BHTE

damage requires a low source voltage and high tissue molecular damage requires high


1N 1S 1M 1L

Case ID

0

100

200

300

400

500

600

700

Ru

ntim

e (

s)

Laplacian

BHTE

Arrhenius

(a) Average runtime per case

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

300

350

400

450

500

550

Maxim

um

targ

et te

mpera

ture

(K

)

SN7

SN10

SN20

SN30

CN25

MN2K30

T = 333.15K (60C)

(b) Maximum target temperature (Case 1N)

Figure 5.11: Average runtimes and maximum target temperature

source voltage (Table 5.12). Finally, our framework indicates the use of a single needle

for targets up to 15 cm3 and multiple needles for larger targets (Figure 5.16). Needle

configurations that do not attain full coverage are not recommended for treatment.

Our multiple needle placement methodology is unable to find non-intersecting needle

positions for smaller tumors using longer multiple needles (e.g., MN3K30, MN3K40 for

Case 1N), and hence no TDO computations were performed for such cases. However,

for Case 3N, NOO-Kmeans could not find non-interesting needle positions for MN3K40,

and none of the other needle configurations were able to obtain 100% target coverage.

In such circumstances, we increase the target size by adding margins to obtain needle

positions and perform TDO analysis for the original target.

When NOO is performed using the IP models, TDO is performed for fewer needle

types than convex models, since IP rejects several needle types due to their intersection

with OAR or critical structures, or due to infeasible multi-needle solutions. Similar

to TDO performed on convex needle placements, a longer needle length causes larger

lesion volumes (Figure 5.17), and consequently higher target (Figure 5.18) and OAR

(Figure 5.19) damage volumes. Similarly, BHTE target coverage occurs at lower voltage

values, while ATDM and percentage damage models require higher voltage values (Figure

5.18). Since needle positions and orientations are discretized, they may not correspond


2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

2

4

6

8

10

12

Lesio

n V

olu

me (

cubic

mm

) T

>=

60C

×10 4

SN7

SN10

SN20

SN30

CN25

MN2K30

(a) BHTE

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1

2

3

4

5

6

7

8

9

Lesio

n V

olu

me (

cubic

mm

) Ω

>=

1

×10 4

SN7

SN10

SN20

SN30

CN25

MN2K30

(b) ATDM

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1

2

3

4

5

6

7

8

9

Lesio

n V

olu

me (

cubic

mm

) D

63

×10 4

SN7

SN10

SN20

SN30

CN25

MN2K30

(c) D63

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1

2

3

4

5

6

7

8

9

Lesio

n V

olu

me (

cubic

mm

) D

70

×10 4

SN7

SN10

SN20

SN30

CN25

MN2K30

(d) D70

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1

2

3

4

5

6

7

8

Lesio

n V

olu

me (

cubic

mm

) D

80

×10 4

SN7

SN10

SN20

SN30

CN25

MN2K30

(e) D80

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1

2

3

4

5

6

7

8

Lesio

n V

olu

me (

cubic

mm

) D

95

×10 4

SN7

SN10

SN20

SN30

CN25

MN2K30

(f) D95

Figure 5.12: Lesion volumes (Case 1N)


2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100P

erc

ent

targ

et

with T

>=

60C

SN7

SN10

SN20

SN30

CN25

MN2K30

(a) BHTE

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Perc

en

t ta

rge

t w

ith

Ω >

= 1

SN7

SN10

SN20

SN30

CN25

MN2K30

(b) ATDM

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D6

3 d

am

ag

e

SN7

SN10

SN20

SN30

CN25

MN2K30

(c) D63

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D7

0 d

am

ag

e

SN7

SN10

SN20

SN30

CN25

MN2K30

(d) D70

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D8

0 d

am

ag

e

SN7

SN10

SN20

SN30

CN25

MN2K30

(e) D80

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D9

5 d

am

ag

e

SN7

SN10

SN20

SN30

CN25

MN2K30

(f) D95

Figure 5.13: Percent target coverage (Case 1N)


2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35P

erc

ent O

AR

with T

>=

60C

SN7

SN10

SN20

SN30

CN25

MN2K30

(a) BHTE

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with

Ω >

= 1

SN7

SN10

SN20

SN30

CN25

MN2K30

(b) ATDM

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

63 d

am

age

SN7

SN10

SN20

SN30

CN25

MN2K30

(c) D63

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

63 d

am

age

SN7

SN10

SN20

SN30

CN25

MN2K30

(d) D70

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

63 d

am

age

SN7

SN10

SN20

SN30

CN25

MN2K30

(e) D80

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

95 d

am

age

SN7

SN10

SN20

SN30

CN25

MN2K30

(f) D95

Figure 5.14: Percent OAR coverage (Case 1N)


2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

0.5

1

1.5

2

2.5

3T

10

0

60

SN7

SN10

SN20

SN30

CN25

MN2K30

(a) BHTE

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

0.5

1

1.5

2

2.5

3

Ω1

00

1

SN7

SN10

SN20

SN30

CN25

MN2K30

(b) ATDM

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

0.5

1

1.5

2

2.5

3

τ1

00

63

SN7

SN10

SN20

SN30

CN25

MN2K30

(c) D63

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

0.5

1

1.5

2

2.5

3

τ1

00

70

SN7

SN10

SN20

SN30

CN25

MN2K30

(d) D70

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

0.5

1

1.5

2

2.5

3

τ1

00

80

SN7

SN10

SN20

SN30

CN25

MN2K30

(e) D80

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

0.5

1

1.5

2

2.5

3

τ1

00

95

SN7

SN10

SN20

SN30

CN25

MN2K30

(f) D95

Figure 5.15: Treatment times, rounded to the closest minute, based on full coverage(Case 1N)


Table 5.12: Recommended needle configurations (Algorithm 7)

Case ID Damage model Needle type Volts (V) OAR damage (%)

1N

BHTE SN7 20.00 1.07ATDM SN7 25.00 0.91D63 SN10 20.00 0.69D70 SN7 25.00 0.84D80 SN10 22.50 0.89D95 SN7 27.50 0.82

2N

BHTE SN20 22.50 4.31ATDM SN20 27.50 3.63D63 SN20 27.50 3.63D70 MN2K30 12.50 4.22D80 MN2K30 12.50 3.87D95 SN30 25.00 4.60

3N

BHTE MN2K30 27.50 6.03ATDM MN3K30 30.00 8.85D63 MN3K30 30.00 8.85D70 MN3K30 30.00 8.57D80 MN3K30 30.00 8.15D95 - - -

0 10 20 30 40 50 60 70 80

Target volumes (cubic cm)

SN7

SN10

SN20

SN30

CN25

MN2K30

MN3K30

MN3K40

Ne

ed

le t

yp

es

Figure 5.16: Recommended needle configuration for BHTE damage model. The voltageis indicated by the star size.


to geometric centroids or shapes of the target clusters. Hence, higher voltages might be

required than our non-trajectory methods to achieve similar lesion volumes resulting in

less OAR coverage.

5.2.4 Discussion

Due to the lack of standards based on either conformity or OAR sparing, complexity

of optimization models, and difference in data sets and needle types used, it is difficult

to draw direct comparisons with existing simultaneous models. In ablation planning,

simultaneous optimization provides the benefit of needle placement by simultaneously

computing thermal damage without any assumptions on ablation shape. Due to the

inherent non-linear nature of ablation, simultaneous optimization methods, which solve

PDEs as constraints with needle position and orientation as the only variables, are only

able to produce locally optimal solutions. They must be tailored to needle type as well

as ablation modality, thereby restricting their clinical viability. Trajectory planning is

difficult to incorporate in such models, and due to long runtimes, experiments on multiple

source voltage (or power) selection is not tractable. Further, due to the mathematical

complexity of ablation optimization models, it is difficult to comment on the quality

of previous studies’ optimal solutions and none of these studies comment on optimality

gaps.

Inexact methods use geometric approximations and unconstrained linear models to

obtain single or multiple needle placements. These methods do not perform thermal dose

simulations and target coverage is determined by assuming a fixed ablation volume, typ-

ically based on vendor specifications. While fast optimal solutions are obtained due to

model simplicities, the treatment accuracies cannot be determined as thermal lesions can

change due to local tissue interactions. Our approach decomposes needle placement and

thermal damage computation which results in inherent sub-optimal solutions. Unlike

inexact methods, we do not assume an ablation volume except when trajectory plan-


2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

2000

4000

6000

8000

10000

12000

14000

Le

sio

n V

olu

me

(cu

bic

mm

) T

>=

60

C

7 mm

10 mm

(a) BHTE

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1000

2000

3000

4000

5000

6000

7000

8000

Le

sio

n V

olu

me

(cu

bic

mm

) Ω

>=

1

7 mm

10 mm

(b) ATDM

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1000

2000

3000

4000

5000

6000

7000

8000

Le

sio

n V

olu

me

(cu

bic

mm

) D

63

7 mm

10 mm

(c) D63

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1000

2000

3000

4000

5000

6000

7000

Le

sio

n V

olu

me

(cu

bic

mm

) D

70

7 mm

10 mm

(d) D70

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1000

2000

3000

4000

5000

6000

7000

Le

sio

n V

olu

me

(cu

bic

mm

) D

80

7 mm

10 mm

(e) D80

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

1000

2000

3000

4000

5000

6000

Le

sio

n V

olu

me

(cu

bic

mm

) D

95

7 mm

10 mm

(f) D95

Figure 5.17: Lesion volumes with trajectory planning (Case 1N)


2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100P

erc

ent

targ

et

with T

>=

60C

7 mm

10 mm

(a) BHTE

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Perc

en

t ta

rge

t w

ith

Ω >

= 1

7 mm

10 mm

(b) ATDM

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D6

3 d

am

ag

e

7 mm

10 mm

(c) D63

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D7

0 d

am

ag

e

7 mm

10 mm

(d) D70

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D8

0 d

am

ag

e

7 mm

10 mm

(e) D80

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

10

20

30

40

50

60

70

80

90

100

Pe

rce

nt

targ

et

with

D9

5 d

am

ag

e

7 mm

10 mm

(f) D95

Figure 5.18: Percent target coverage with trajectory planning (Case 1N)


2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35P

erc

ent O

AR

with T

>=

60C

7 mm

10 mm

(a) BHTE

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with

Ω >

= 1

7 mm

10 mm

(b) ATDM

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

63 d

am

age

7 mm

10 mm

(c) D63

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

63 d

am

age

7 mm

10 mm

(d) D70

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

63 d

am

age

7 mm

10 mm

(e) D80

2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

Source Voltages (V)

0

5

10

15

20

25

30

35

Perc

ent O

AR

with D

95 d

am

age

7 mm

10 mm

(f) D95

Figure 5.19: Percent OAR coverage with trajectory planning (Case 1N)


ning is performed. Further, any assumptions on geometric shapes or ablation sizes are

lifted when thermal damage is computed. Despite these shortcomings, our approach pro-

vides several benefits over existing methods, including computational advantage, thermal

damage analysis, flexibility towards ablation modality, ability to incorporate trajectory

planning, and ability to include multiple needle types.

Typically, a good cancer treatment plan will provide a full conformal target coverage

with maximum OAR sparing. However, unlike radiation, rigidity of heat deposition

makes it difficult to control the shape or spread of ablation. If full target coverage is

the only necessary requirement, then any needle position that achieves this goal is an

acceptable solution. However, it is obvious that some needle positions are better than

others. For instance, a needle that is larger than target radius placed closer to target

boundary may provide full coverage but is less desirable than one closer to the center

of the target. This choice can be attributed to better target thermal dose, coverage of

microscopic tumor particles surrounding the target, and less OAR damage. Existing

models do not provide information on OAR sparing and use different data sets for any

comparative analysis, and we did not find any standard in the literature to evaluate

quality of an RFA treatment based on either OAR sparing or target conformity.

In radiotherapy, two conformity indices are commonly used to quantify the treatment

quality: classic and Paddick conformity indices (Section 5.1.2). The classic index (CI) is

the ratio of target volume to the ablated volume. Even if the target and ablation volumes

do not overlap, a treatment plan with a low CI value can be labeled as a good conformal

treatment plan. The Paddick index (as well as Dice similarity coefficient) overcomes

this drawback by incorporating information regarding the target and ablation volume

overlaps, and measures the similarity between these sets. However, these metrics do not

provide information regarding uniformity or evenness of the dose distribution around the

target. Assuming full target coverage, to quantify the quality of an RFA treatment plan,

a metric that determines the uniformity or evenness of thermal distribution around the


target should be developed. For instance, the proximity between the geometric centers of

the target and the ablated lesion combined with their volumes can help determine how

evenly the thermal doses are deposited around the target. If the distance between the

geometric centers is large (> dg), then the thermal dose is unevenly spread around the

target, potentially indicating poor coverage of microscopic tumor extensions. Further,

over- or under-ablation may be found if the ratio between target and ablated volumes is

not within a threshold range, [κmin, κmax]. The value of κmax must be < 1 since a larger

value indicates incomplete target coverage. A treatment can be quantified as good only

if the conditions on concentricity as well as volume ratio are satisfied. Finally, these

threshold values depend on the target site, where more (e.g., breast, prostate) or less

(e.g., liver) OAR-sparing is recommended, and targets’ relative position within that site

(e.g., boundary or internal of the site), and therefore, must be clinically provided.

Intuitively, for a single needle placement, the needle position will correspond to the

centroid of the target and its orientation will correspond to the shape of the target.

This hypothesis has been previously validated through experiments using simultaneous

optimization [2]. Our fast convex NOO model, NOO-MVCE, delivers similar solutions.

For multiple needle placement, we provide detailed methodological explanation absent in

previous work using a similar approach [15].

The thermal dose for the largest target (Case 3L) is computed in <20 minutes, which

is a significant improvement over the 1-2 hours reported by simultaneous optimization [2,

15]. We can easily extend our work to other ablation modalities by solving a different set

of PDEs, e.g., Maxwell’s equations for MWA, which can be difficult in PDE-constrained

systems. Further, new needle types can be seamlessly added in the NOO stage without

affecting the TDO methodology.

Similar to simultaneous optimization methods, we assume a fixed treatment time,

which in our case is 20 minutes. This conservative longer treatment time gives us enough

simulation data to analyze the treatment quality while ensuring maximum target cov-


erage. However, for the recommended needle configurations, full coverage is achieved

within the first few minutes. Since tissues eventually reach thermal equilibrium, treat-

ment time does not significantly affect the treatment quality, unlike radiation treatments.

Gradual heat deposition with longer treatment times ensures larger ablation volumes, and

therefore improved coverage of microscopic tumor particles.

Most RFA treatment planning systems that incorporate trajectory planning focus

on single needle placement [30, 56, 57, 59] or employ sequential integer programming

techniques for multiple needle placement that result in suboptimal solutions [52, 66].

None of these approaches incorporate thermal dose computations. The non-linearity

of PDE-constrained systems makes it difficult to incorporate trajectory planning for

multiple needle placements [30]. We extend our two-stage RFA planning framework for

single, clustered, and multiple needle placement by incorporating trajectory planning

[35] and for the first time present a full RFA planning framework that incorporates

trajectory planing as well as thermal dose computations. In this work, we use IP models,

solved to global optimality, to identify the best needle positions for single or multiple

needle placement based on geometric approximations. We are still able to attain full

target coverage, albeit at higher source voltage than our non-trajectory planning TDO.

Further, trajectory planning disregards large needle types for smaller targets as well

as invalid needle trajectories, unlike our non-trajectory planning work where all needle

lengths were explored for a target. However, trajectory planning will fail to find any

needle placements when the target diameter is smaller than the smallest needle tip length,

but we can overcome this drawback by artificially increasing the target size. Thus, our

trajectory planning model provides a realistic advantage to our convex models.

In the trajectory planning framework, we assume an ablation radius for each indi-

vidual needle inserted and approximate it to an ellipse. However, vendor specifications

provide ablation radii for multiple needles where needles are placed parallely and oper-

ated simultaneously on a porcine liver. The shape of the lesion is unclear when multiple


Table 5.13: Description of deflection scenarios. rmin = minimum deflection radius (mm);rmax = maximum deflection radius (mm); |M| = total scenarios; |M| = total sampledscenarios; |Γ| = total deflected angles; nP = number of processors

Case IDTranslational scenarios Combined scenarios

rmin rmax |M| |M| nP rmin rmax |Γ| |M| |M| nP

1N 0 4 213 213 48 0 4 8 1704 1704 801S 0 4 257 257 48 0 4 8 2056 2056 801M 0 4 257 257 48 0 4 8 2056 2056 801L 0 4 257 257 48 0 4 8 2056 2056 80

2N 0 4 257 257 80 0 4 8 2056 2056 802S 0 2 1089 1089 80 0 0 26 676 676 802M 0 2 1089 1089 80 0 0 26 676 676 802L 0 2 1089 759 48 0 0 26 676 676 48

needles are placed non-parallely due to a lack of clinical experiments. Thus, the NOO

stage may incorrectly estimate the ablation radius and consequently the target cover-

age. The target coverage can be determined by thermal dose simulations as well as by

enforcing a minimum number of ablations based on clinical experience.

5.3 Needle deflection results

Base case needle positions and orientations are obtained from model NOO-MVCE for

single needle (Case 1) and NOO-Kmeans for multiple needle ablation (Cases 2 and 3).

The thermal distributions are computed on our in-house cluster with 256 AMD Opteron™

Processor 4332 HE, 3 GHz CPUs with a maximum of 120 MATLAB 2015b workers. The

number of processors used for computation depend on target size, number of scenarios,

and worker availability (Table 5.13).

A large deflection radius creates a large number of scenarios. To analyze the impact

of 2-4 mm imaging errors on target damage, a maximum of 4 mm deflection radius is used

for single needle ablation and 2 mm for multiple needle ablation for both translational

and combined scenarios. The number of scenarios generated and executed depends on the


Table 5.14: Base case needle configuration for needle deflection analysis

Case ID Needle type Source voltage (V)

1N SN7 251S SN20 17.51M SN20 22.51L SN30 25

2N SN20 27.52S MN2K30 152M MN2K30 152L MN2K30 20

number of ablations, target size, and deflection distance (Table 5.13). For multiple needle

deflection, we sample scenarios with 50% or more likelihood of occurrence. Although we

report target damages for BHTE as well as D63 damage model, the base case uses the

D63 damage model without trajectory planning (Table 5.14).

5.3.1 Single needle deflection

Figure 5.20 shows target damage across all Case 1 single needle ablation scenarios using

BHTE and D63 damage models. Better D63 target coverage (≥ 94%) is seen for Cases

1M and 1L across translational scenarios; while for Cases 1N and 1S worst case D63 tar-

get coverage loss is up to 20% and 15%, respectively. Similar observations can be made

for combined deflection scenarios, where deflection angles Γ = ±6.5,±10,±15,±20

are used. However, target coverage loss is higher, up to 30%, than translational scenarios,

especially for smaller targets. Finally, D63 damage model ablates the target slower than

BHTE and the base case needle configuration is based on D63 damage model. There-

fore, when using BHTE damage model, majority of Case 1 translational and combined

scenarios show ≥ 95% target coverage. Hence, we analyze D63 damage for individual

scenarios.

As the percentage of target coverage decreases, the percentage of OAR coverage in-

creases (Figure 5.21). For a fixed deflection angle, similar trends are found in combined


Translational deflection Combined deflection

D63

79 84 89 94 990

10

20

30

40

50

60

70

80

90

100

Percent of target with at least 63% cellular damage

Pe

rce

nt

of

sce

na

rio

s

1N1S1M1L

70 75 80 85 90 95 1000

10

20

30

40

50

60

70

80

90

100


Pe

rce

nt

of

sce

na

rio

s

2S2M2L

BHTE

79 84 89 94 990

10

20

30

40

50

60

70

80

90

100

Percent of target with at least 60C temperature

Pe

rce

nt

of

sce

na

rio

s

1N1S1M1L

70 75 80 85 90 95 1000

10

20

30

40

50

60

70

80

90

100


Pe

rce

nt

of

sce

na

rio

s

2S2M2L

Figure 5.20: Single needle ablation: Target damage across scenarios


85 90 95 1000.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Percent target covered

Perc

ent O

AR

covere

d

Deflected scenariosBase case scenario

(a) Translational scenarios

65 70 75 80 85 90 95 1000.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Percent targed covered

Perc

ent O

AR

covere

d

±6.5°

±10°

±15°

±20°

Base case scenario

(b) Combined scenarios

Figure 5.21: Single needle ablation: Target v. OAR D63 damage (Case 1N)

−300 −200 −100 0 100 200 3000

0.5

1

1.5

2

2.5

3

3.5

4

Position vector angle (deg)

De

fle

ctio

n f

rom

ba

se

ca

se

(m

m)

2

4

6

8

10

12

14

16

18

Percent target uncovered


−300 −200 −100 0 100 200 3000

0.5

1

1.5

2

2.5

3

3.5

4

Position vector angle (deg)

De

fle

ctio

n f

rom

ba

se

ca

se

(m

m)

5

10

15

20

Percent target uncovered

(b) Combined scenarios (γ = ±20)

Figure 5.22: Single needle ablation: D63 target coverage (Case 1N)

deflection scenarios. However, an increase in deflection angle does not necessarily indicate

an increase in target coverage loss or OAR coverage.

Figure 5.22 shows the effect of deflection distance on target coverage, where position

vector angle refers to ω1 and ω2 in Figure 5.23, which uniquely differentiates multiple

deflected centers at same distance. An increase in deflection distance decreases the target

coverage and is indicated by a lighter shade for Case 1N translational scenarios (Figure

5.21(a)). Similar trends can be seen for Case 1N combined scenarios for γ = ±20 with an

additional 2-3% decrease in target coverage than translation scenarios (Figure 5.21(b)).


−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x−axis

y−

axis

uv

ω1

ω2

Figure 5.23: Position vector angles

5.3.2 Multiple needle deflection

For multiple needle ablation, in order to understand the effect of deflection in a needle’s

base orientation, we perform combined deflection analysis with a fixed base case needle

center and deflection angles Γ ∈ [−60, 60] in the increments of 5. For both the damage

models, multiple needle ablations give ≥99% and ≥97% damage across all translational

and combined deflection scenarios, respectively (Figure 5.24). Thus, unlike single needle

ablation, small translational or rotational deflections do not affect the treatment quality

of multiple needle ablations due to their ability to create large thermal lesions. As inter-

needle distance increases, which is the Euclidean distance between the needle centers, the

lesion volumes also increases (Figure 5.25). Consequently, there is a full target coverage

but an increase in OAR coverage (Figure 5.26). However, for a fixed needle distance,

thermal lesions vary in size when inter-needle angles, which is the angle in degrees between

the two needles, change from their base orientations (Figure 5.25). Consequently, similar

pattern is seen in OAR coverage but almost full target coverage is maintained (Figure

5.27).


Translational deflection Combined deflection

D63

98 99 1000

10

20

30

40

50

60

70

80

90

100


Pe

rce

nt

of

sce

na

rio

s

2S2M2L

97 98 99 1000

10

20

30

40

50

60

70

80

90

100


Pe

rce

nt

of

sce

na

rio

s

2S2M2L

BHTE

98 99 1000

10

20

30

40

50

60

70

80

90

100


Pe

rce

nt

of

sce

na

rio

s

2S2M2L

97 98 99 1000

10

20

30

40

50

60

70

80

90

100


Pe

rce

nt

of

sce

na

rio

s

2S2M2L

Figure 5.24: Multi-needle ablation: Target damage across scenarios

6 8 10 12 14 163

3.5

4

4.5

5

5.5

6x 10

4

Inter−needle distance (mm)

Lesio

n v

olu

mes (

cubic

mm

)


−60 −40 −20 0 20 40 603

3.5

4

4.5

5

5.5

6

6.5x 10

4

Inter−needle angles (deg)

Lesio

n v

olu

mes (

cubic

mm

)

(b) Combined scenarios

Figure 5.25: Multi-needle ablation: Lesion volumes and inter-needle parameters (Case2S)


7 8 9 10 11 12 13 14 15 1699

99.1

99.2

99.3

99.4

99.5

99.6

99.7

99.8

99.9

100


Pe

rce

nt

targ

et

co

ve

red


(a) Target

7 8 9 10 11 12 13 14 15 16

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5


Perc

ent O

AR

covere

d


(b) OAR

Figure 5.26: Multi-needle ablation: Needle distance and D63 damage for translationalscenes (Case 2S)

−60 −40 −20 0 20 40 6099.4

99.5

99.6

99.7

99.8

99.9

100

Inter−needle angle (deg)

Pe

rce

nt

targ

et

co

ve

red


(a) Target

−60 −40 −20 0 20 40 603.5

4

4.5

5

5.5

6

6.5

7

Inter−needle angle (deg)

Perc

ent O

AR

covere

d


(b) OAR

Figure 5.27: Multi-needle ablation: Needle angles and D63 damage for combined scenes(Case 2S)


5.3.3 Discussion

Although inaccuracies in needle placement may be common in ablation treatment, this

work empirically explores its effect on treatment outcome for the first time. For single

needle ablation, results show that OAR coverage increases as target coverage decreases

indicating a shift in thermal lesion. Further, lower needle configurations, i.e., short

needle tip lengths, number of needles, and source voltage, are used for small targets and

therefore translational deflections >2 mm can significantly reduce target coverage up

to 20%. Further, if a needle orientation changes more than 5 along with translational

deflection, target coverage loss can potentially increase up to 30%. However, for larger

targets, higher needle configuration is used and hence better target coverage is obtained

(Cases 1M and 1L). Therefore, an increase in source voltage might mitigate the effect of

deflection for smaller targets, albeit at a cost of higher OAR coverage. Thus, based on

our preliminary empirical analysis, we can hypothesize that any larger deflection distance

or angles will decrease target coverage significantly.

Multiple needle ablations are used to treat large targets due to their ability to produce

large thermal lesions. As needles deflect further from each other, the thermal lesion size

increases causing higher OAR coverage. In our simulations, we explored translational

deflections up to 2 mm and therefore the largest inter-needle distance change is 4 mm. We

obtain full target coverage for all the scenarios. However, for inter-needle distances > 4

mm, it is possible that some central target voxels are unablated. Since small translational

deflection does not affect target coverage for multiple needle ablations, we performed

experiments to understand the effect of rotational deflections with a fixed inter-needle

distance. As needles deflect further from their base orientations, the overall thermal

lesion changes, reflected in the OAR coverage. However, full target coverage is seen since

needles’ centers are fixed at their base center. Thus, full target coverage can be obtained

despite small deflections, and changes to input voltage or power might be able to improve

any central target coverage loss for larger inter-needle distances. Further experiments,


especially for larger inter-needle distances, are required to determine a threshold when

multiple needles result in multiple thermal lesions instead of a single large ablation zone.

Proper trajectory planning can help avoid deflections due to dense organs like ribs,

however, there can be other causes for deflection, e.g., changes in tissue geometry due to

needle insertions or involuntary patient movement due to breathing. Clinical studies are

recommended to understand various deflection causes to help reduce the high recurrence

rate of ablation. Finally, robust treatments that provide full target coverage and a good

quality treatment despite needle deflections should be generated, e.g., a source voltage

or power that provides full target coverage for worst-case needle placement.

Chapter 6

Conclusions and future work

This work presents a major step forward in systematic treatment planning of small-

to-medium inoperable hepatic tumors using RFA. We presented a novel mathematical

RFA framework that separates needle placement and thermal dose computation, which

improves the computational tractability of developing a plan by eliminating iterative

computation of thermal dose. Our framework extends to other ablation modalities (e.g.,

MWA), considers treatment planning for eight different needle types with the ability to

accommodate other needle types (e.g., umbrella-shaped needles), and analyzes target and

OAR damage using multiple damage models. Full target coverage is obtained for three

clinical liver cases with four margin sizes. Finally, our methods return the best needle

configuration based on full target and minimum OAR coverage.

Our NOO algorithms obtain needle positions and orientations for single, multiple,

and clustered needle scenarios with and without trajectory planning. We presented fast

sophisticated convex optimization methods that use ellipse- and sphere-based geometric

approximations for needle placement without trajectory planning. A variation of set cover

integer program is used to select multiple pairwise valid ellipses for full target coverage,

where each ellipse represents a valid needle path, whose tractability and scalability is

improved using row generation techniques. For single and multiple needle ablations,

92

Chapter 6. Conclusions and future work 93

unpredictable translational and rotational needle deflections can occur during treatment

and we explore their effect on target retention. Thus, for the first time we incorporate

needle trajectories with TDO, perform damage analysis on both target and OAR using

multiple damage models, and demonstrate the impact of needle deflection on target and

OAR coverage.

Typically, image guidance is used for RFA needle placement and multiple reinser-

tions are required until desired placement is found. The resulting inaccuracies in needle

placement are a cause for local recurrence especially for large targets that require multiple

ablations. Further, needle placements are restricted due to physical and visual challenges.

Although, RFA treatments are less than 30 minutes, these added difficulties can increase

the overall treatment time. Our work addresses these challenges by providing thermal

dose simulations on mathematically obtained single and multiple needle positions. We

also help decision makers to determine input parameters for a desired damage model and

their impact on target kill and OAR sparing. Finally, our work addresses the inaccuracies

in image guidance that can significantly affect target coverage, recommending a need for

better understanding of deflection causes.

Our trajectory planning algorithm can be further improved by incorporating other

path selection criteria like unablated target, path length, proximity to critical structures

like veins, and total ablations. The use of commercial PDE solvers that include RF mod-

ules [16] can enhance the quality of thermal dose simulations, and therefore treatment

plans, since several physiological, thermal, and electrical processes (e.g, change in tissue

thermal properties with temperature change) are not captured by the mathematical sim-

ulations presented here. Additional experiments are recommended to analyze the impact

of large translational and rotational deflections, especially for multiple needle ablations

due to their ability to create larger lesions than single needles. Simulations that analyze

whether the effect of deflections can be mitigated by a change in input parameters, e.g.,

source voltage or power, will help physicians determine if needle reinsertions are required

Chapter 6. Conclusions and future work 94

during an intra-operative procedure. Finally, before clinical use, our methods require

significant thermal dose validations, e.g., in porcine tissue. Such validations can be chal-

lenging as they can only be performed ex-vivo as it is difficult to obtain in-vivo thermal

distributions without MRI imaging, which interfere with the metallic ablation needles,

and thermometers can only provide local tissue temperatures.

Bibliography

[1] Muneeb Ahmed and S Nahum Goldberg. Radiofrequency tissue ablation: principles

and techniques. In Radiofrequency Ablation for Cancer, pages 3–28. Springer, 2004.

[2] Inga Altrogge, Tobias Preusser, Tim Kroger, Christof Buskens, Philippe L. Pereira,

Diethard Schmidt, and Heinz-Otto Peitgen. Multiscale optimization of the probe

placement for radiofrequency ablation. Academic radiology, 14(11):1310–24, 2007.

ISSN 1076-6332. doi: 10.1016/j.acra.2007.07.016. URL http://linkinghub.

elsevier.com/retrieve/pii/S1076633207004448.

[3] John Amanatides and Andrew Woo. A fast voxel traversal algorithm for ray tracing.

In Eurographics, pages 3–10, 1987.

[4] Claire Baegert, Caroline Villard, Pascal Schreck, and Luc Soler. Multi-criteria tra-

jectory planning for hepatic radiofrequency ablation. In Proceedings of the 10th in-

ternational conference on Medical image computing and computer-assisted interven-

tion, MICCAI’07, pages 676–684, Berlin, Heidelberg, 2007. Springer-Verlag. ISBN

3-540-75758-9, 978-3-540-75758-0. URL http://dl.acm.org/citation.cfm?id=

1775835.1775926.

[5] Jeffrey Bax, Derek Cool, Lori Gardi, Kerry Knight, David Smith, Jacques Montreuil,

Shi Sherebrin, Cesare Romagnoli, and Aaron Fenster. Mechanically assisted 3d

ultrasound guided prostate biopsy system. Medical Physics, 35(12):5397–5410, 2008.

95

http://linkinghub.elsevier.com/retrieve/pii/S1076633207004448


http://dl.acm.org/citation.cfm?id=1775835.1775926

http://dl.acm.org/citation.cfm?id=1775835.1775926

BIBLIOGRAPHY 96

ISSN 2473-4209. doi: 10.1118/1.3002415. URL http://dx.doi.org/10.1118/1.

3002415.

[6] Neil Bhardwaj, Andrew Strickland, Fateh Ahmad, Ashley Dennison, and David

Lloyd. Liver ablation techniques: A review. Surgical Endoscopy, 24:254–265, 2010.

ISSN 0930-2794. doi: 10.1007/s00464-009-0590-4. URL http://dx.doi.org/10.

1007/s00464-009-0590-4.

[7] Lorenz T. Biegler, Omar Ghattas, Matthias Heinkenschloss, and Bart van Bloe-

men Waanders. Large-scale pde-constrained optimization: An introduction. In

Lorenz T. Biegler, Matthias Heinkenschloss, Omar Ghattas, and Bart van Bloe-

men Waanders, editors, Large-Scale PDE-Constrained Optimization, pages 3–13,

Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. ISBN 978-3-642-55508-4.

[8] Harry Blum. A transformation for extracting new descriptors of shape. Models for

the perception of speech and visual form, 19(5):362–380, 1967.

[9] Stephen Boyd and Lieven Vandenberghe. Semidefinite programming. SIAM Review,

38(1):49–95, 1996.

[10] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge Univer-

sity Press, 2004.

[11] Torste Butz, Simon Warfield, Kemal Tuncali, Stuart Silverman, Eric van Sonnen-

berg, Ferenc Jolesz, and Ron Kikinis. Pre- and intra-operative planning and simu-

lation of percutaneous tumor ablation. International Conference on Medical Image

Computing and Computer Assisted Intervention, 3:317–326, 10 2000.

[12] Isaac Chang and Uyen Nguyen. Thermal modeling of lesion growth with

radiofrequency ablation devices. BioMedical Engineering OnLine, 3(1):27,

2004. ISSN 1475-925X. doi: 10.1186/1475-925X-3-27. URL http://www.

biomedical-engineering-online.com/content/3/1/27.

http://dx.doi.org/10.1118/1.3002415

http://dx.doi.org/10.1118/1.3002415

http://dx.doi.org/10.1007/s00464-009-0590-4

http://dx.doi.org/10.1007/s00464-009-0590-4

http://www.biomedical-engineering-online.com/content/3/1/27

http://www.biomedical-engineering-online.com/content/3/1/27

BIBLIOGRAPHY 97

[13] Issac Chang and Bryan B. Beard. Precision test apparatus for evaluating the heating

pattern of radiofrequency ablation devices. Medical Engineering & Physics, 24:633–

640, 2002.

[14] Chun-Cheng R. Chen, Michael Miga, and Robert Galloway. Optimizing needle

placement in treatment planning of radiofrequency ablation. In Medical Imaging

2006: Visualization, Image-Guided Procedures, and Display, volume 6141, pages

632–638, 2006. ISBN 0277786X. doi: 10.1117/12.654983. URL http://link.aip.

org/link/PSISDG/v6141/i1/p614124/s1&Agg=doi.

[15] Chun-Cheng R. Chen, Michael I. Miga, and Robert L. Galloway. Optimizing elec-

trode placement using finite-element models in radiofrequency ablation treatment

planning. IEEE Transactions on Biomedical Engineering, 56(2):237–45, 2009. ISSN

0018-9294. doi: 10.1109/TBME.2008.2010383. URL http://ieeexplore.ieee.

org/lpdocs/epic03/wrapper.htm?arnumber=4694126.

[16] COMSOL Inc. Microwave and RF design software. http://www.comsol.com/

rf-module, 2017. [Online; accessed 5-January-2017].

[17] Covidien. http://www.covidien.com/surgical/products/ablation-systems/

cool-tip-rf-ablation-system-e-series-electrodes, 2015. [Online; accessed

05-August-2015].

[18] CVX Research. CVX: Matlab software for disciplined convex programming, version

2.0 beta. http://cvxr.com/cvx, September 2012.

[19] Lee R. Dice. Measures of the amount of ecologic association between species. Ecology,

26(3):pp. 297–302, 1945. ISSN 00129658. URL http://www.jstor.org/stable/

1932409.

[20] Emad Ebbini, Shin-Ichiro. Umemura, Mohammed Ibbini, and Charles A. Cain. A

cylindrical-section ultrasound phased-array applicator for hyperthermia cancer ther-

http://link.aip.org/link/PSISDG/v6141/i1/p614124/s1&Agg=doi

http://link.aip.org/link/PSISDG/v6141/i1/p614124/s1&Agg=doi

http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4694126

http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=4694126

http://www.comsol.com/rf-module

http://www.comsol.com/rf-module

http://www.covidien.com/surgical/products/ablation-systems/cool-tip-rf-ablation-system-e-series-electrodes

http://www.covidien.com/surgical/products/ablation-systems/cool-tip-rf-ablation-system-e-series-electrodes

http://cvxr.com/cvx

http://www.jstor.org/stable/1932409

http://www.jstor.org/stable/1932409

BIBLIOGRAPHY 98

apy. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 35

(5):561–572, 1988. ISSN 0885-3010. doi: 10.1109/58.8034.

[21] Jeffrey Eggleston and Wolf W. Von Maltzahn. The Biomedical Engineering Hand-

book, Second Edition. CRC Press LLC, Boca Raton,USA, 1999.

[22] Aaron Fenster, Donal B Downey, and H Neale Cardinal. Three-dimensional ultra-

sound imaging. Physics in medicine & biology, 46(5):R67, 2001.

[23] Michael Ferris and David M Shepard. Optimization of gamma knife radiosurgery.

Discrete Mathematical Problems with Medical Applications, 55:27–44, 2000.

[24] Michael Ferris, Jinho Lim, and David Shepard. An optimization approach for

radiosurgery treatment planning. SIAM Journal on Optimization, 13(3):921–937,

2002. doi: 10.1137/S105262340139745X. URL http://dx.doi.org/10.1137/

S105262340139745X.

[25] Kimia Ghobadi, Hamid Ghaffari, Dionne Aleman, David Jaffray, and Mark Ruschin.

Automated treatment planning for a dedicated multi-source intracranial radio-

surgery treatment unit using projected gradient and grassfire algorithms. Medical

Physics, 39(6):3134–3141, 2012. doi: 10.1118/1.4709603.

[26] Kimia Ghobadi, Hamid Ghaffari, Dionne Aleman, David Jaffray, and Mark Ruschin.

Automated treatment planning for a dedicated multi-source intra-cranial radio-

surgery treatment unit accounting for overlapping structures and dose homogeneity.

Medical Physics, 40(9), 2013.

[27] S Nahum Goldberg, Scott Gazelle, and Peter Mueller. Thermal ablation therapy

for focal malignancy: a unified approach to underlying principles, techniques, and

diagnostic imaging guidance. American Journal of Roentgenology, 174(2):323–331,

2000. URL http://www.ajronline.org/content/174/2/323.short.

http://dx.doi.org/10.1137/S105262340139745X

http://dx.doi.org/10.1137/S105262340139745X

http://www.ajronline.org/content/174/2/323.short

BIBLIOGRAPHY 99

[28] Michael C. Grant and Stephen Boyd. Graph implementations for nonsmooth convex

programs. In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in

Learning and Control, Lecture Notes in Control and Information Sciences, pages

95–110. Springer-Verlag Limited, 2008. http://stanford.edu/~boyd/graph_dcp.

html.

[29] Inc. Gurobi Optimization. Gurobi optimizer reference manual, 2014. URL http:

//www.gurobi.com.

[30] Sabrina Haase, Philipp Suss, Jan Schwientek, Katrin Teichert, and Tobias Preusser.

Radiofrequency ablation planning: An application of semi-infinite modelling tech-

niques. European Journal of Operational Research, 218(3):856–864, 2012. ISSN

03772217. doi: 10.1016/j.ejor.2011.12.014. URL http://linkinghub.elsevier.

com/retrieve/pii/S0377221711010915.

[31] FC Henriques. Studies of thermal injury v. the predictability and significance of

thermally induced rate processes leading to irreversible epidermal injury. American

Journal of Pathology, 43:489–502, 1947.

[32] FC Henriques and AR Moritz. Studies of thermal injury in the conduction of heat

to and through skin and the temperatures attained therein: A theoretical and ex-

perimental investigation. American Journal of Pathology, 23:531–549, 1947.

[33] Huang-Wen Huang, Tzu-Ching Shih, and Chihng-Tsung Liauh. Predicting effects

of blood flow rate and size of vessels in a vasculature on hyperthermia treatments

using computer simulation. BioMedical Engineering OnLine, 9(1):18, Mar 2010. doi:

10.1186/1475-925X-9-18. URL https://doi.org/10.1186/1475-925X-9-18.

[34] Tommy Knoos, Ingrid Kristensen, and Per Nilsson. Volumetric and dosimetric

evaluation of radiation treatment plans: radiation conformity index. Interna-

tional Journal of Radiation Oncology*Biology*Physics, 42(5):1169 – 1176, 1998.

http://stanford.edu/~boyd/graph_dcp.html

http://stanford.edu/~boyd/graph_dcp.html

http://www.gurobi.com

http://www.gurobi.com



https://doi.org/10.1186/1475-925X-9-18

BIBLIOGRAPHY 100

ISSN 0360-3016. doi: http://dx.doi.org/10.1016/S0360-3016(98)00239-9. URL

http://www.sciencedirect.com/science/article/pii/S0360301698002399.

[35] Shefali Kulkarni-Thaker, Dionne Aleman, and Aaron Fenster. Inverse planning for

radiofrequency ablation in cancer therapy using multiple damage models. Submitted,

2018.

[36] Shaoli Liu, Zeyang Xia, Jianhua Liu, Jing Xu, He Ren, Tong Lu, and Xiangdong

Yang. Automatic multiple-needle surgical planning of robotic-assisted microwave

coagulation in large liver tumor therapy. PloS one, 11(3):e0149482, 2016.

[37] Stuart Lloyd. Least squares quantization in PCM. IEEE transactions on information

theory, 28(2):129–137, 1982.

[38] Medtronic. Cool-tip RF Ablation System E Series Electrodes and Ac-

cessories. http://www.medtronic.com/covidien/products/ablation-systems/

cool-tip-rf-ablation-system-e-series-electrodes, 2016. [Online; accessed

04-October-2016].

[39] AR Moritz. Studies of thermal injury III. The pathology and pathogenesis of cuta-

neous burns: An experimental study. American Journal of Pathology, 23:915–934,

1947.

[40] AR Moritz and FC Henriques. Studies of thermal injury II. The relative importance

of time and surface temperature in the causation of cutaneous burns. American

Journal of Pathology, 23:695–720, 1947.

[41] S. Mulier, Y. Ni, Y. Miao, A. Rosiere, A. Khoury, G. Marchal, and Michel L. Size and

geometry of hepatic radiofrequency lesions. Europian Journal of Surgical Oncology,

29:867–878, 2003.

http://www.sciencedirect.com/science/article/pii/S0360301698002399

http://www.medtronic.com/covidien/products/ablation-systems/cool-tip-rf-ablation-system-e-series-electrodes

http://www.medtronic.com/covidien/products/ablation-systems/cool-tip-rf-ablation-system-e-series-electrodes

BIBLIOGRAPHY 101

[42] Laurent Mundeleer, David Wikler, Thierry Leloup, V. Lucidi, Vincent Donckier,

and Nadine Warzee. Computer-assisted needle positioning for liver tumour radiofre-

quency ablation. International Journal of Medical Robotics, 5(4):458–64, 2009. ISSN

1478-5951. doi: 10.1002/rcs.278. URL http://doi.wiley.com/10.1002/rcs.v5%

3A4.

[43] Hamid Neshat, Derek W. Cool, Kevin Barker, Lori Gardi, Nirmal Kakani, and

Aaron Fenster. A 3d ultrasound scanning system for image guided liver interventions.

Medical Physics, 40(11):112903–n/a, 2013. ISSN 2473-4209. doi: 10.1118/1.4824326.

URL http://dx.doi.org/10.1118/1.4824326. 112903.

[44] NIST. NIST reference on constants units and uncertainity. http://physics.nist.

gov/cgi-bin/cuu/Value?r, 2015. [Online; accessed 12-January-2013].

[45] Ann P O’Rourke, Dieter Haemmerich, Punit Prakash, Mark C Converse, David M

Mahvi, and John G Webster. Current status of liver tumor ablation devices. Expert

Review of Medical Devices, 4(4):523–537, 2007. doi: 10.1586/17434440.4.4.523. URL

http://dx.doi.org/10.1586/17434440.4.4.523.

[46] Ian Paddick. A simple scoring ratio to index the conformity of radiosurgi-

cal treatment plans. Journal of Neurosurgery, 93(supplement 3):219–222, 2000.

doi: 10.3171/jns.2000.93.supplement3.0219. URL http://thejns.org/doi/abs/

10.3171/jns.2000.93.supplement%203.0219. PMID: 11143252.

[47] John Pearce. Relationship between Arrhenius models of thermal damage and the

CEM 43 thermal dose. In Proceedings of SPIE, Energy-based Treatment of Tissue

and Assessment V, volume 7181, pages 718104–718104–15, 2009. doi: 10.1117/12.

807999. URL +http://dx.doi.org/10.1117/12.807999.

[48] Adrian Scott Pearson, Francesco Izzo, R.Y. Declan Fleming, Lee Ellis, Paolo Delrio,

Mark Roh, Jennifer Granchi, and Steven Curley. Intraoperative radiofrequency

http://doi.wiley.com/10.1002/rcs.v5%3A4

http://doi.wiley.com/10.1002/rcs.v5%3A4

http://dx.doi.org/10.1118/1.4824326

http://physics.nist.gov/cgi-bin/cuu/Value?r

http://physics.nist.gov/cgi-bin/cuu/Value?r

http://dx.doi.org/10.1586/17434440.4.4.523

http://thejns.org/doi/abs/10.3171/jns.2000.93.supplement%203.0219

http://thejns.org/doi/abs/10.3171/jns.2000.93.supplement%203.0219

+ http://dx.doi.org/10.1117/12.807999

BIBLIOGRAPHY 102

ablation or cryoablation for hepatic malignancies. The American journal of surgery,

178(6):592–598, 1999.

[49] Jiming Peng and Yu Wei. Approximating k-means-type clustering via semidefinite

programming. SIAM Journal on Optimization, 18(1):186–205, 2007.

[50] Punit Prakash, Vasant A. Salgaonkar, E. Clif Burdette, and Chris J. Diederich. Mul-

tiple applicator hepatic ablation with interstitial ultrasound devices: Theoretical and

experimental investigation. Medical Physics, 39(12):7338–7349, 2012. ISSN 2473-

4209. doi: 10.1118/1.4765459. URL http://dx.doi.org/10.1118/1.4765459.

[51] Hansjorg Rempp, Andreas Boss, Thomas Helmberger, and Philippe Pereira. The

current role of minimally invasive therapies in the management of liver tumors.

Abdominal imaging, 36(6):635–647, 2011.

[52] Hongliang Ren, Enrique Campos-Nanez, Ziv Yaniv, Filip Banovac, Hernan Abeledo,

Nobuhiko Hata, and Kevin Cleary. Treatment planning and image guidance for

radiofrequency ablation of large tumors. IEEE journal of biomedical and health

informatics, 18(3):920–928, 2014.

[53] H. Edwin Romeijn, Ravindra K. Ahuja, James F. Dempsey, and Arvind Kumar. A

new linear programming approach to radiation therapy treatment planning prob-

lems. Operations Research, 54(2):201–216, 2006. doi: 10.1287/opre.1050.0261. URL

http://dx.doi.org/10.1287/opre.1050.0261.

[54] H.Edwin Romeijn and James Dempsey. Intensity modulated radiation therapy treat-

ment plan optimization. TOP, 16(2):215–243, 2008. ISSN 1134-5764. doi: 10.1007/

s11750-008-0064-1. URL http://dx.doi.org/10.1007/s11750-008-0064-1.

[55] Stephen Sapareto and William Dewey. Thermal dose determination in cancer ther-

apy. International Journal of Radiation Oncology Biology Physics, 10(6):787 – 800,

http://dx.doi.org/10.1118/1.4765459

http://dx.doi.org/10.1287/opre.1050.0261

http://dx.doi.org/10.1007/s11750-008-0064-1

BIBLIOGRAPHY 103

1984. doi: 10.1016/0360-3016(84)90379-1. URL http://www.sciencedirect.com/

science/article/pii/0360301684903791.

[56] Christian Schumann, Jennifer Bieberstein, Christoph Trumm, Diethard Schmidt,

Philipp Bruners, Matthias Niethammer, Ralf T. Hoffmann, Andreas H. Mahnken,

Philippe L. Pereira, and Heinz-Otto Peitgen. Fast automatic path proposal com-

putation for hepatic needle placement. In Medical Imaging 2010: Visualization,

Image-Guided Procedures, and Modeling, volume 7625 of Proceedings in SPIE,

pages 76251J–76251J–10, 2010. ISBN 0277786X. doi: 10.1117/12.844186. URL

http://link.aip.org/link/PSISDG/v7625/i1/p76251J/s1&Agg=doi.

[57] Christian Schumann, Christian Rieder, Sabrina Haase, Katrin Teichert, Philipp

Suss, Peter Isfort, Philipp Bruners, and Tobias Preusser. Interactive multi-criteria

planning for radiofrequency ablation. International journal of computer assisted

radiology and surgery, 10(6):879–889, 2015.

[58] Hans-Joachim Schwarzmaier, Anna N. Yaroslavsky, Ilya V .and Yaroslavsky,

Volkhard Fiedler, Frank Ulrich, and Thomas Kahn. Treatment planning for mri-

guided laser-induced interstitial thermotherapy of brain tumors—the role of blood

perfusion. Journal of Magnetic Resonance Imaging, 8(1):121–127, 1998. ISSN 1522-

2586. doi: 10.1002/jmri.1880080124. URL http://dx.doi.org/10.1002/jmri.

1880080124.

[59] Alexander Seitel, Markus Engel, Christof M. Sommer, Boris A. Radeleff, Caroline

Essert-Villard, Claire Baegert, Markus Fangerau, Klaus H. Fritzsche, Kwong Yung,

Hans-Peter Meinzer, and Lena Maier-Hein. Computer-assisted trajectory plan-

ning for percutaneous needle insertions. Medical Physics, 38(6):3246–, 2011. ISSN

00942405. doi: 10.1118/1.3590374. URL http://link.aip.org/link/MPHYA6/

v38/i6/p3246/s1&Agg=doi.

http://www.sciencedirect.com/science/article/pii/0360301684903791

http://www.sciencedirect.com/science/article/pii/0360301684903791

http://link.aip.org/link/PSISDG/v7625/i1/p76251J/s1&Agg=doi

http://dx.doi.org/10.1002/jmri.1880080124

http://dx.doi.org/10.1002/jmri.1880080124

http://link.aip.org/link/MPHYA6/v38/i6/p3246/s1&Agg=doi

http://link.aip.org/link/MPHYA6/v38/i6/p3246/s1&Agg=doi

BIBLIOGRAPHY 104

[60] Walter A Strauss. Partial differential equations: An introduction. New York, 1992.

[61] Sharon Thomsen and John A. Pearce. Thermal damage and rate processes in bio-

logic tissues. In Ashley J. Welch and Martin J.C. Gemert, editors, Optical-Thermal

Response of Laser-Irradiated Tissue, pages 487–549. Springer Netherlands, 2011.

URL http://dx.doi.org/10.1007/978-90-481-8831-4_13. 10.1007/978-90-481-

8831-4 13.

[62] Supan Tungjitkusolmun, Tyler T. Staelin, Dieter Haemmerich, Tsai Jang-Zern, Cao

Hong, John G. Webster, Fred T. Lee Jr., David M. Mahvi, and Vicken R. Vorperian.

Three-dimensional finite-element analyses for radio-frequency hepatic tumor abla-

tion. IEEE Transactions on Biomedical Engineering, 49(1):3 –9, Jan 2002. ISSN

0018-9294. doi: 10.1109/10.972834.

[63] Caroline Villard, Luc Soler, and Afshin Gangi. Radiofrequency ablation of hep-

atic tumors: simulation, planning, and contribution of virtual reality and hap-

tics. Computer Methods in Biomechanics and Biomedical Engineering, 8(4):215–

227, 2005. ISSN 1025-5842. doi: 10.1080/10255840500289988. URL http:

//www.tandfonline.com/doi/abs/10.1080/10255840500289988.

[64] WHO. Media center. http://www.who.int/mediacentre/factsheets/fs297/en/,

2012. [Online; accessed 13-December-2012].

[65] Eugene Wissler. Pennes’ 1948 paper revisited. Journal of Applied Physiology, 85(1):

35–41, 1998. URL http://jap.physiology.org/content/85/1/35.abstract.

[66] Ziv Yaniv, Patrick Cheng, Emmanuel Wilson, Teo Popa, David Lindisch, Enrique

Campos-Nanez, Hernan Abeledo, Vance Watson, Kevin Cleary, and Filip Banovac.

Needle-based interventions with the image-guided surgery toolkit (igstk): from phan-

toms to clinical trials. IEEE Transactions on Biomedical Engineering, 57(4):922–933,

2010.

http://dx.doi.org/10.1007/978-90-481-8831-4_13

http://www.tandfonline.com/doi/abs/10.1080/10255840500289988

http://www.tandfonline.com/doi/abs/10.1080/10255840500289988

http://www.who.int/mediacentre/factsheets/fs297/en/

http://jap.physiology.org/content/85/1/35.abstract

BIBLIOGRAPHY 105

[67] Hui Zhang, Filip Banovac, Stella Munuo, Enrique Campos-Nanez, Hernan Abeledo,

and Kevin Cleary. Treatment planning and image guidance for radiofrequency ab-

lation of liver tumors. In Proceedings of SPIE, Medical Imaging 2007: Visualization

and Image-Guided Procedures, volume 6509, pages 650922–650922–10, 2007. doi:

10.1117/12.711489. URL http://dx.doi.org/10.1117/12.711489.

http://dx.doi.org/10.1117/12.711489

List of acronyms

Acronym Description

ATDM Arrhenius thermal damage modelBHTE Bioheat transfer equationCEM Cummulative equivalent minuresCI Classic conformity indexDSC Dice similarity coefficientFC Feasibility checkHIFU High frequency focused ultrasoundMVCE Minimum volume covering ellipsoidMVCS Minimum volume covering sphereMWA Microwave ablationNOO Needle orientation optimizationOAR Organs-at-riskPDE Partial differential equationPI Paddick indexRFA Radiofrequency ablationRFA-SCP Set cover problem for radiofrequency ablationRMP Reduced master problemSAR Specific absorption rateSQP Sequential quadratic programingTDO Thermal dose optimization

106

by Shefali Kulkarni-Thaker - University of Toronto T-Space · 2018-07-18 · I would also like to...

Documents

Transcript of by Shefali Kulkarni-Thaker - University of Toronto T-Space · 2018-07-18 · I would also like to...