OPTIMIZATION METHODS IN INTENSITY ...

OPTIMIZATION METHODS IN INTENSITY MODULATED RADIATION THERAPY TREATMENT PLANNING

By DIONNE M. ALEMAN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 1

c 2007 Dionne M. Aleman

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To my ever-patient wife Nancy, and to my father Roberto, who, if not for the shortcomings of current cancer treatments, might still be with us today

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ACKNOWLEDGMENTS Many thanks to Nancy Huang, Christopher Fox and Bart Lynch for so helpfully and happily explaining the physics of medical physics to me on a wide range of topics, even when those topics are not relevant to my own research. This work was supported in part by the NSF Alliances for Graduate Education and the Professoriate, the NSF Graduate Research Fellowship and NSF grant DMI-0457394.

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TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.1 1.2

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12 13 13 14 15 15 16 16 17 19 19

FLUENCE MAP OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1 2.2 2.3 2.4 2.5

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2.6

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Intensity Modulated Radiation Therapy Treatment Planning Dissertation Summary . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fluence Map Optimization . . . . . . . . . . . . . . . 1.2.2 Beam Orientation Optimization . . . . . . . . . . . . 1.2.3 Fractionation . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Modeling the Dose Deposition of a Beam . . . . . . . Contribution Summary . . . . . . . . . . . . . . . . . . . . . 1.3.1 Fluence map optimization . . . . . . . . . . . . . . . 1.3.2 Beam Orientation Optimization . . . . . . . . . . . . 1.3.3 Fractionation . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Modeling the Dose Deposition of a Beam . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . Literature Review . . . . . . . . . . . . . . . . . . Model Formulation . . . . . . . . . . . . . . . . . Spatial Considerations . . . . . . . . . . . . . . . A Primal-Dual Interior Point Algorithm for FMO 2.5.1 Primal-Dual Interior Point Algorithm . . . 2.5.2 Hessian Approximations . . . . . . . . . . 2.5.2.1 Single Hessian Approximation . . 2.5.2.2 BFGS Hessian Update . . . . . . 2.5.3 Insignificant Beamlets . . . . . . . . . . . . 2.5.4 Warm Start . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 How Small of a Duality Gap is Necessary? 2.6.2 Computational Results . . . . . . . . . . . 2.6.3 Clinical Results . . . . . . . . . . . . . . . 2.6.4 Spatial Coefficient Results . . . . . . . . . 2.6.5 Warm Start Results . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . .

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BEAM ORIENTATION OPTIMIZATION . . . . . . . . . . . . . . . . . . . . .

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3.1 3.2 3.3 3.4 3.5 3.6

46 47 48 50 52 54 55 58 59 59 61 69 69 69 70 70 71 72 72 73 75 75 75 76 77 79 79 79 80 81 83 84 85 88 89 89 91 92 92 95

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3.8

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed-Integer Model Formulation . . . . . . . . . . . . . . . . . . . Beam Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . A Response Surface Approach to BOO . . . . . . . . . . . . . . . . 3.6.1 Overview of Response Surfaces . . . . . . . . . . . . . . . . . 3.6.2 Determining the Next Observation . . . . . . . . . . . . . . . 3.6.2.1 Maximizing the expected improvement . . . . . . . 3.6.2.2 Obtaining an upper bound on the uncertainty . . . 3.6.2.3 Branch-and-Bound . . . . . . . . . . . . . . . . . . 3.6.3 Method of Obtaining the Next Observation . . . . . . . . . . Neighborhood Search . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Neighborhood Search Approaches . . . . . . . . . . . . . . . 3.7.3 A Deterministic Neighborhood Search Method for BOO . . . 3.7.3.1 Neighborhood Definition . . . . . . . . . . . . . . . 3.7.3.2 Neighbor Selection . . . . . . . . . . . . . . . . . . 3.7.3.3 Implementation . . . . . . . . . . . . . . . . . . . . 3.7.4 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . 3.7.4.1 Neighborhood Definition . . . . . . . . . . . . . . . 3.7.4.2 Neighbor Selection . . . . . . . . . . . . . . . . . . 3.7.4.3 Implementation . . . . . . . . . . . . . . . . . . . . 3.7.4.4 Convergence . . . . . . . . . . . . . . . . . . . . . . 3.7.5 A New Neighborhood Structure . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Evaluating Plan Quality . . . . . . . . . . . . . . . . . . . . 3.8.1.1 Target coverage . . . . . . . . . . . . . . . . . . . . 3.8.1.2 Critical structure sparing . . . . . . . . . . . . . . 3.8.2 Response Surface Method Results . . . . . . . . . . . . . . . 3.8.2.1 Proof of concept . . . . . . . . . . . . . . . . . . . 3.8.2.2 Adding a non-coplanar beam to a coplanar solution 3.8.2.3 Clinical results . . . . . . . . . . . . . . . . . . . . 3.8.3 Neighborhood Search Method Results . . . . . . . . . . . . . 3.8.3.1 Add/Drop algorithm results . . . . . . . . . . . . . 3.8.3.2 Simulated Annealing results . . . . . . . . . . . . . 3.8.3.3 Clinical results . . . . . . . . . . . . . . . . . . . . Conclusions and Future Directions . . . . . . . . . . . . . . . . . . 3.9.1 Response Surface Conclusions . . . . . . . . . . . . . . . . . 3.9.2 Neighborhood Search Conclusions . . . . . . . . . . . . . . .

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FRACTIONATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3

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Introduction . . . . . . . . . . . . . Model Formulation . . . . . . . . . Results . . . . . . . . . . . . . . . . 4.3.1 Computational Results . . . 4.3.2 Clinical Results . . . . . . . 4.3.3 Spatial Coefficient Results . Conclusions and Future Directions

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96 96 97 100 101 102 103 111

A MONTE CARLO METHOD FOR MODELING DOSE DEPOSITION . . . . 120 5.1 5.2 5.3

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5.5 5.6

Introduction . . . . . . . . . . . . . . . Monte Carlo Engine . . . . . . . . . . Dose Distribution of a Beamlet . . . . 5.3.1 Depth-Dose Curve . . . . . . . . 5.3.2 Lateral Penumbra . . . . . . . . Methodology to Model a Beamlet . . . 5.4.1 Modeling the Depth-Dose Curve 5.4.2 Modeling the Lateral Penumbra Results . . . . . . . . . . . . . . . . . . Conclusions and Future Directions . .

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120 121 121 122 123 124 125 128 132 138

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

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LIST OF TABLES Table

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2-1 Average run times for 5-beam treatment plans . . . . . . . . . . . . . . . . . . .

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2-2 FMO value obtained using = 0.001 . . . . . . . . . . . . . . . . . . . . . . . .

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2-3 Comparison of duality gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2-4 Performance measures of interior point method warm starts . . . . . . . . . . .

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2-5 Performance measures of projected gradient method warm starts . . . . . . . . .

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3-1 Sparing criteria varies for each critical structure . . . . . . . . . . . . . . . . . .

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3-2 Sizes of test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3-3 Minimum FMO value obtained and time required to obtain it . . . . . . . . . .

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3-4 Target coverage achieved by the treatment plans . . . . . . . . . . . . . . . . . .

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3-5 Percentage of plans in which an organ is spared . . . . . . . . . . . . . . . . . .

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3-6 Definitions of implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4-1 Case sizes and run times using identical algorithm and weighting parameters . . 102 4-2 Sparing criteria varies for each critical structure . . . . . . . . . . . . . . . . . . 103 5-1 Computation times in minutes of Monte Carlo simulations . . . . . . . . . . . . 132 5-2 Computation times for dose distribution fits . . . . . . . . . . . . . . . . . . . . 134 5-3 Variation of fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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LIST OF FIGURES Figure

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2-1 Progression of duality gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2-2 Dose received by targets as a function of the duality gap . . . . . . . . . . . . .

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2-3 Dose received by saliva glands as a function of the duality gap . . . . . . . . . .

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2-4 Quality of DVHs for various duality gaps . . . . . . . . . . . . . . . . . . . . . .

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2-5 The spatial coefficients used for two cases . . . . . . . . . . . . . . . . . . . . .

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2-6 Comparison of spatial and non-spatial treatment plans . . . . . . . . . . . . . .

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2-7 Comparison of spatial and non-spatial treatment plans . . . . . . . . . . . . . .

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3-1 A linear accelerator and the available movements . . . . . . . . . . . . . . . . .

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3-2 FMO value as a function of two angles . . . . . . . . . . . . . . . . . . . . . . .

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3-3 Initial regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3-4 Partitioning a region into subregions . . . . . . . . . . . . . . . . . . . . . . . .

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3-5 Accounting for symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3-6 The flip neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3-7 Selection probabilities in Nh (θ) and NhF (θ) . . . . . . . . . . . . . . . . . . . .

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3-8 Proof of concept results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3-9 Comparison of response surface, Add/Drop and equi-spaced targets . . . . . . .

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3-10 Comparison of response surface, Add/Drop and equi-spaced targets . . . . . . .

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3-11 Add/Drop and simulated annealing comparison of FMO convergence . . . . . .

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3-12 Comparison of Add/Drop and 7-beam equi-spaced plans . . . . . . . . . . . . .

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3-13 Comparison of simulated annealing and 7-beam equi-spaced plans . . . . . . . .

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4-1 Target DVHs, saliva DVHs and axial slices in Fractions 1 and 2 . . . . . . . . . 104 4-2 Target DVHs, saliva DVHs and axial slices in Fractions 1 and 2 . . . . . . . . . 105 4-3 Target DVHs, saliva DVHs and axial slices in Fractions 1 and 2 . . . . . . . . . 106 4-4 Target DVHs, saliva DVHs and axial slices in Fractions 1 and 2 . . . . . . . . . 107 4-5 Target DVHs, saliva DVHs and axial slices in Fractions 1 and 2 . . . . . . . . . 108

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4-6 Target DVHs, saliva DVHs and axial slices in Fractions 1 and 2 . . . . . . . . . 109 4-7 Target DVHs, saliva DVHs and axial slices in Fractions 1 and 2 . . . . . . . . . 110 4-8 DVHs and axial slices in Fractions 1 and 2 using spatial coefficients . . . . . . . 112 4-9 DVHs and axial slices in Fractions 1 and 2 using spatial coefficients . . . . . . . 113 4-10 DVHs and axial slices in Fractions 1 and 2 using spatial coefficients . . . . . . . 114 4-11 DVHs and axial slices in Fractions 1 and 2 using spatial coefficients . . . . . . . 115 4-12 DVHs and axial slices in Fractions 1 and 2 using spatial coefficients . . . . . . . 116 4-13 DVHs and axial slices in Fractions 1 and 2 using spatial coefficients . . . . . . . 117 4-14 DVHs and axial slices in Fractions 1 and 2 using spatial coefficients . . . . . . . 118 5-1 Dose distribution of a single beamlet in various tissues . . . . . . . . . . . . . . 122 5-2 Colorwash of the lateral penumbra of a finite sized pencil beam . . . . . . . . . 124 5-3 Plot of the lateral penumbra of a finite sized pencil beam . . . . . . . . . . . . . 125 5-4 Observed depth-dose curve in water for several histories . . . . . . . . . . . . . . 126 5-5 Polynomial fits of several histories . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5-6 Variation of polynomial fit as function of degree . . . . . . . . . . . . . . . . . . 128 5-7 An error function and an error function pair . . . . . . . . . . . . . . . . . . . . 129 5-8 Lateral penumbra for several numbers of Monte Carlo histories . . . . . . . . . . 130 5-9 Error function fits of several histories . . . . . . . . . . . . . . . . . . . . . . . . 131 5-10 Error function pairs summed to approximate a beamlet in water . . . . . . . . . 135 5-11 Depth-dose curves in muscle tissue. . . . . . . . . . . . . . . . . . . . . . . . . . 135 5-12 Lateral penumbra curves in muscle tissue. . . . . . . . . . . . . . . . . . . . . . 136 5-13 Depth-dose curves in lung tissue. . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5-14 Lateral penumbra curves in lung tissue. . . . . . . . . . . . . . . . . . . . . . . . 137 5-15 Depth-dose curves in heterogeneous muscle and lung tissue.

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5-16 Variation of fits as a function of number of histories . . . . . . . . . . . . . . . . 139

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMIZATION METHODS IN INTENSITY MODULATED RADIATION THERAPY TREATMENT PLANNING By Dionne M. Aleman December 2007 Chair: H. Edwin Romeijn Major: Industrial and Systems Engineering The design of a treatment plan for intensity modulated radiation therapy a mathematical programming problem which is not yet satisfactorily solved. Current techniques include dividing the problem into several subproblems, which are then solved sequentially. My research addresses several of these subproblems, particularly, beam orientation optimization (BOO), fluence map optimization (FMO) and fractionation. The integration of the BOO and FMO subproblems is considered, as well as improved techniques to model the dose deposition of a beamlet.

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CHAPTER 1 INTRODUCTION 1.1

Intensity Modulated Radiation Therapy Treatment Planning

Every year, approximately 1.4 million people in the United States alone are newly diagnosed with cancer (American Cancer Society, [1]). More than half of these patients will receive some form of radiation therapy (Murphy et al. [2], Perez and Brady [3]), and approximately half of these patients may significantly benefit from conformal radiation therapy (Steel [4]). During this therapy, beams of radiation pass through a patient, thereby killing both cancerous and normal cells. Although some patients die of their disease despite sophisticated treatment methods, many patients may suffer unpleasant side effects as a result of the radiation therapy which may severely detract from the patient’s quality of life. Thus, the radiation treatment must be carefully planned so that a clinically prescribed dose is delivered to targets containing cancerous cells so that the cancer will be eradicated. Simultaneously, a small enough dose must be delivered to the nearby organs and tissues (called critical structures) so that they may survive the treatment. This is achieved by irradiating the patient using several beams sent at different orientations spaced around the patient so that the intersection of these beams includes the targets, which thus receive the highest radiation dose, whereas the critical structures receive radiation from some, but not all, beams and may thus be spared. Currently, a technique called intensity modulated radiation therapy (IMRT) is considered to be the most effective radiation therapy for many forms of cancer. The problem of designing an IMRT treatment plan for an individual patient is a large-scale mathematical programming problem that is not yet solved satisfactorily. Current treatment planning systems decompose the planning problem into several stages, and the corresponding subproblems are solved sequentially. These subproblems include determining the number and orientation of the beams of radiation, the radiation dose

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distribution of each beam and the decomposition of a single treatment plan into several smaller fractions. This work addresses the integration of the beam orientation optimization (BOO) and fluence map optimization (FMO) subproblems based on a convex formulation of the latter and associated efficient algorithms for solving it, an approach which has not received much attention in previous studies. The fractionation problem, the problem of dividing a single treatment plan into the 35 treatments (fractions) the patient will actually receive, is also addressed. Also, the problem of modeling the dose deposition of a beam is also considered. 1.2

Dissertation Summary

In IMRT, each beam is modeled as a collection of hundreds of small beamlets, the fluences of which can be controlled individually. These fluence values are known as a fluence map, and optimization of these fluences given a fixed set of beams is known as fluence map optimization. The optimal solution value of the FMO problem quantifies the quality of the treatment plan, where quality means the ability of the plan to deliver the prescribed radiation dose to the specified target structures while sparing critical structures by ensuring that they receive an acceptably low amount of radiation. Thus, the quality of a set of beams can be measured by the optimal solution of the FMO problem performed with those beams. Thus, the problem of selecting the best directions from which to deliver radiation to the patient (the BOO problem) is based on the treatment plan quality indicated by the optimal solution value to the corresponding FMO problem. 1.2.1

Fluence Map Optimization

One of the most popular subproblems of the intensity modulated radiation therapy (IMRT) treatment planning problem is the fluence map optimization (FMO) problem. In IMRT, each beam of radiation can be discretized in hundreds of smaller beamlets, the radiation intensities (fluences) of which can be modulated independently of the other beamlets. For a given set of beams, the beamlet fluences can greatly influence the quality of the treatment plan, that is, the ability of the treatment to deposit the prescibed amount

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of dose to cancerous target structures while simultaneously delivering a small enough dose to critical structures so that they may continue to function after the treatment. These fluence values are known as a fluence map, and optimization of these fluences given a fixed set of beams is known as fluence map optimization. Because the fluences of the beamlets can drastically affect the quality of the treatment plan it is critical to obtain good fluence maps for radiation delivery. As the FMO problem is one of the most popular subproblems in IMRT optimization, it has been extensively studied in the literature. Several problem structures and algorithms to solve various models are presented in many studies. 1.2.2

Beam Orientation Optimization

In a typical head-and-neck treatment plan, radiation beams are delivered from 5-9 nominally-spaced coplanar orientations around the patient. These coplanar orientations are obtained from rotating the gantry only. Several components of a linear accelerator can rotate and translate to achieve more orientations than those obtained from rotating the gantry. The available orientations consist of the orientations obtained from rotation of the gantry, collimator and couch, as well as the three translation directions of the couch. Beam orientation optimization (BOO) is the problem of selecting from the available beam orientations the best set to use in delivering a treatment plan. Given a fixed set of beams, different fluence maps (radiation intensities of beamlets) yield treatment plans with different qualities. Therefore, the quality of an optimized fluence map should be considered when selecting a set of beam orientations to use in a treatment plan. Optimal fluence maps may be difficult to obtain depending on the FMO model. Thus, it is common in the literature for scoring approximations and other heuristics to be used to estimate the quality of a beam solution. Regardless of the objective function used in the BOO problem, the problem is fundamentally nonlinear as the physics of dose deposition change with direction. Because nonlinear programming problems are difficult to solve, most approaches to the BOO

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problem rely on global search algorithms to obtain a solution, which may or may not be optimal. 1.2.3

Fractionation

An important subproblem related to the FMO problem which has not yet received much attention is the fractionation problem. Rather than deliver an entire treatment plan in one session, a treatment plan is divided into several sessions, called fractions. This is done to take advantage of the fact that normal, healthy cells recover faster from the radiation than cancerous cells. To obtain the treatment plans for the fractions, in practice, a single FMO treatment plan is developed and then divided into the desired number of fractions, usually around 35. This division of a treatment plan is a non-trivial task, as the target voxels, geometric cubes of tissue, must receive 1.8-2.0 Gy of radiation in each fraction. With a single IMRT treatment plan, it is practically impossible to devise a constant dose-per-fraction delivery technique because only a single FMO problem is solved to obtain the treatment plan, which is then simply divided into a number of daily fractions. If a single plan is optimized to deliver doses to multiple target-dose levels, then the dose per fraction delivered to each target must change in the ratio of a given dose level to the maximum dose level. For example, say PTV1 has a prescription dose of 70 Gy, PTV2 has a prescription dose of 50 Gy, and the number of fractions is 35. If a single treatment plan is divided among the 35 fractions, then PTV1 will receive 70/35 = 2.0 Gy in each fraction, but PTV2 will only receive 50/35 = 1.4 Gy, and thus any cancerous cells in PTV2 may not be eradicated by the treatment. Similarly, if only 25 fractions are used in order to ensure that PTV2 receives 2.0 Gy per fraction, then PTV1 receives 70/25 = 2.8 Gy per fraction, well above the desired dose. 1.2.4

Modeling the Dose Deposition of a Beam

The FMO problem is arguably the most significant in determining the quality of the treatment plan. The FMO problem depends heavily on the calculation of dose

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received in each voxel of a patient. This dose is typically approximated by assuming a linear relationship with the radiation intensities of the beamlets delivering the radiation. Although this approximation is accepted as satisfactory, it is not truly accurate. The dose in a voxel is determined by the paths the photons in the radiation beams follow through the patient. Some photons may collide with particles inside the patient and scatter in any direction, while others may collide with particles and be absorbed. Still other photons may pass entirely through the patient with no collisions. Due to the unpredictable nature of the radiation beam inside the patient, the dose received in a voxel can only be accurately obtained through Monte Carlo simulations. A simple linear relationship is assumed between total dose and beamlet fluences and is commonly accepted as a satisfactory dose approximation in IMRT optimization. Errors of as much as 30% have been reported for photon beams near tissue inhomogeneities (Ma et al. [5]). For IMRT optimization, particularly with advent of image-guided IMRT (IGIMRT), or 4D IMRT, the FMO problem must be solved extremely quickly to create real-time treatment plans. Thus, the speed of the FMO problem is paramount. Lengthy Monte Carlo simulation can provide an accurate measure of the dose deposited in a voxel, but this technique is time intensive and impractical for clinical use and particularly for treatment planning optimization. 1.3 1.3.1

Contribution Summary

Fluence map optimization

Nonlinear functions to approximate biological behavior and desired dose distributions are common in the previously proposed FMO models in the literature, as are mixed-integer programming models. These models can be difficult and computationally expensive to solve. To make the FMO problem more tractable, we employ a model with a convex objective function and linear constraints. This desirable structure allows our model to be solved quickly and to optimality with the primal-dual interior point algorithm we have developed specifically for this problem.

16

One of the greatest benefits of an interior point algorithm is that a globally optimal solution can be found for many problem structures, and in particular, convex problem structures. As our FMO model is convex, the interior point algorithm can locate the globally optimal solution to within a specified duality gap. While there are other algorithms that can theoretically return a globally optimal solution to a convex problem (and many algorithms that cannot), interior point methods have the advantage of providing a known duality gap and generally fast computation times. Because the duality gap is known in each iteration, the user can make knowledgeable trade-offs between computation time and solution optimality without having to guess how far from the optimum the final solution may be. This allows for a scientific comparison of different IMRT delivery techniques as we can solve the different problems to a specific duality gap. Several alterations to the standard primal-dual interior point method were made to improve its performance. Beamlets that are likely to have little or no contribution to the treatment plan are removed a priori and different approximations to the objective function Hessian are tested to save time in calculating the true Hessian in each iteration. The use of warm starts to initialize the interior point method is also examined. The solutions obtained provide quality treatment plans in a clinically feasible amount of time. The incorporation of spatial information into the FMO model is also considered. The probability of tumor metastasis increases with proximity to gross tumor mass. By using the distances of voxels from target structures, the voxels can be weighted according to their importance in the treatment plan. For example, it should be less important to spare saliva gland voxels near a target structure than it should be to spare saliva gland voxels far from a target. The use of spatial coefficients will help the model identify quality treatment plans that will prevent future metastasis. 1.3.2

Beam Orientation Optimization

For head-and-neck cancers, typical IMRT treatment plans use 5-9 equi-spaced coplanar beams. Coplanar beams are those beams obtained from the rotation of only

17

the gantry of the linear accelerator, the machine which delivers radiation beams to the patient. If all other components of the linear accelerator are fixed, the rotation of the gantry sweeps out a set of coplanar beams. The couch can rotate and translate in three dimensions, and the head of the gantry can rotate independently, creating an even larger set of beams. Beams obtained from the movement of more than one component from the linear accelerator are known as non-coplanar beams. Intuitively, one may expect that the number of beams required for a high-quality treatment plan can be reduced, or the quality of the treatment plan for a given number of beams can be improved, if the beam orientations are chosen optimally and/or from a larger set. In particular, we investigate the effect of considering more coplanar or non-coplanar beams. A treatment plan consisting of fewer beams is preferable because the number of beams used in a plan directly affects the length of the actual treatment. If fewer beams are used to treat a patient, then each treatment takes less time and more patients can be treated in a day, which is beneficial from both a clinical and economic perspective. Longer treatment times also allow for more errors due to possible patient motion. We view the BOO problem in IMRT treatment planning as a global optimization problem with expensive objective function evaluations, each of which involves solving a FMO problem. We propose a response surface method that, unlike other approaches, allows for the generation of problem data only for promising beam orientations on-the-fly as the algorithm progresses, enabling the consideration of far more candidate orientations than is currently feasible. Our response surface approach to BOO allows us to develop high quality plans using just four beams for head-and-neck cases, in contrast to the current practice of using 5-9 beams. The response surface method also provides for convergence to the globally optimal solution. We have developed neighborhood search methods to solve our BOO model. One method is simulated annealing, a proper global optimization algorithm, and the other

18

is a local search heuristic designed specifically for the BOO problem. The local search heuristic, which we call the Add/Drop method, returns a locally optimal solution in a small amount of time. The simulated annealing algorithm has the ability to escape local minima, and is theoretically able to return a globally optimal solution given enough time. For each of these algorithms, we have devised a new neighborhood structure based on observations of known optimal BOO solutions compared to the simulated annealing and Add/Drop BOO solutions. This new neighborhood structure provides faster objective function value convergence in both algorithms. 1.3.3

Fractionation

In practice, a single FMO treatment plan is developed and then divided into the number of desired fractions. Dividing a single FMO into multiple treatments is a non-trivial task, owing to the need of maintaining a constant dose-per-fraction to each the target structures, which may have different prescription doses. Therefore, any division of a single FMO plan into multiple fractions can lead to suboptimal treatments. We propose a new method of formulating the fractionation problem which yields optimal fluence maps for each cancerous target structure. These fluence maps can then be easily divided into optimal fractions. The proposed fractionation model is solved using the same primal-dual interior point method presented for the FMO problem. The solutions provide high quality fluence maps for each target, and in a clinically acceptable amount of time. 1.3.4

Modeling the Dose Deposition of a Beam

We propose obtaining a limited number of Monte Carlo histories to obtain a noisy dose distribution which can then be transformed into a very accurate, smooth dose distribution suitable for optimization techniques in a reasonable amount of time. Because the particles in a beamlet scatter in three dimensional space, multiple dose distributions must be considered to satisfactorily model the beamlet’s affect on the patient’s tissue. These distributions arise from the amount of radiation the beamlet

19

deposits as a function of depth (the depth-dose curve), and from the amount of radiation radiating outward from the center of the beamlet (the lateral penumbra). The depth-dose curve is modeled using a high-degree polynomial and the lateral penumbra is modeled as the sum of error functions. The parameters of the error functions are determined using a Levenberg-Marquardt quasi-Newton minimization method. Using these techniques, dose distributions with satisfactory accuracy can be obtained using at least a factor of 10 fewer Monte Carlo histories than would otherwise be required. This can greatly decrease the amount of time required to obtain dose data for beamlets in the FMO problem of IMRT treatment planning without sacrificing accuracy.

20

CHAPTER 2 FLUENCE MAP OPTIMIZATION 2.1

Introduction

IMRT is differentiated from conformal radiation therapy by the dose distributions that can be delivered by each beam. Rather that just delivering a uniform radiation field of radiation, the dose distribution of a beam can be any desired distribution. This ability allows for greater flexibility and accuracy in targeting the target structures while avoiding the critical structures. The dose distribution of a beam is achieved as follows. In IMRT, each beam can be thought of as consisting of several hundred smaller beamlets, each of which can have its own radiation intensity (fluence) independent of its neighbors. By modulating the intensities of these beamlets, any dose distribution can be achieved. Given a fixed set of beams, the optimization of these intensities is called fluence map optimization. 2.2

Literature Review

Because the FMO problem is one of the most studied problems of IMRT, many different approaches have been taken to formulate the FMO problem, based on both “physical” (Bortfeld [6]) and “biological” (Alber and Nusslin [7], Jones and Hoban [8], Kallman et al. [9], Mavroidis et al. [10], Niemierko et al. [11], Niemierko [12], Wu et al. [13, 14]) objective functions and constraints. Linear programming (LP)-based multi-criteria optimization (Hamacher and K¨ ufer [15]) and mixed-integer linear programming (MILP) (Bednarz et al. [16], Ferris et al. [17], Langer et al. [18, 19], Lee et al. [20, 21], Shepard et al. [22]) models have been proposed for FMO. Constraints to enforce various measures of treatment quality are also taken into account in different FMO models. Hamacher and K¨ ufer [15] include the homogeneity of the dose received by the targets as a constraint in their FMO model. Full-volume constraints, which require that the dose in every voxel of a structure be within pre-determined upper and lower bounds, are common for controlling the dose in each structure. Models

21

employing full-volume constraints are found in Bednarz et al. [16], Hamacher and K¨ ufer [15], Lee et al. [20, 21], Romeijn et al. [23] and many others. Models containing partial volume constraints, constraints requiring that dose in only a subset of voxels be within pre-determined upper and/or lower bounds, are also common. Formulations with partial volume constraints are found in Lee et al. [20, 21], Romeijn et al. [23, 24] and Shepard et al. [22]. In addition to varying constraints, there are many competing methods of formulating the FMO objective function to reflect the quality of the treatment plan. Shepard et al. [22] describe several different objective formulations. These formulations include minimizing the sum of doses received at all voxels; minimizing a weighted combination of doses received at each voxel, where the weights depend on the structure in which the voxel resides; and minimizing the deviation of the dose in each voxel from the recommended prescription. Romeijn et al. [25] showed that most of the treatment plan evaluation criteria proposed in the medical physics literature are equivalent to convex penalty function criteria when viewed as a multicriteria optimization problem. For each set of treatment plan evaluation criteria from a very large class, there exists a class of convex penalty functions that produces an identical Pareto efficient frontier. Therefore, a convex penalty function-based approach to evaluating treatment plans is used to investigate the BOO problem. Although this approach could be used in a multicriteria setting, Romeijn et al. [23, 26] suggest that it is possible to quantify a trade-off between the different evaluation criteria that produces high-quality treatment plans for a population of patients, eliminating the need to solve the FMO problem as a multicriteria optimization problem for each individual patient. 2.3

Model Formulation

A convex penalty function-based approach to the FMO model as described in Romeijn et al. [23] is employed to quantify the quality of the treatment plan by appropriately

22

making the trade-off between delivering the prescribed radiation dose to the target structures while sparing the critical structures. Using this approach, the FMO problem can formulated as a quadratic programming problem with linear constraints as follows. Denote the set of all potential beam orientations as B. The structures (both targets and critical structures) are irradiated using a predetermined set of beam angles, denoted θ, where each beam θh ∈ B, h = 1, . . . , k and k is the number of beams in θ. Each beam is decomposed into a rectangular grid of beamlets with m rows and n columns, yielding typically 100-400 beamlets per beam. The position and intensity of all beamlets in a beam can be represented by a vector of values representing the beamlet intensities, called bixels. The set of all bixels in beam θh is denoted by Bθh . The core task in IMRT treatment planning is finding radiation intensities for all beamlets. Denote the total number of structures by S and the number of targets by T . Each structure s is discretized into a finite number vs of volume cubes, known as voxels. Typically, around 350,000 voxels are required to accurately represent the targets and surrounding structures of a head-and-neck cancer site. Because a beamlet must pass through a certain amount of tissue to reach a voxel, the dose received in a voxel from a beamlet may not be the full delivered intensity. Denote Dijs as the dose received by voxel j in structure s from beamlet i at unit intensity. The Dijs values are known as dose deposition coefficients. Let xi denote the intensity of bixel i. This brings us to the following expression for the dose zjs received by voxel j in structure s: zjs =

k X X

Dijs xi

j = 1, . . . , vs , s = 1, . . . , S

h=1 i∈Bθh

Although the goal of IMRT treatment planning is to control the dose received by each structure, if hard constraints are imposed on the amount of dose received by each structure because such a solution may not exist. In some cases, it may be necessary to sacrifice organs in order to treat targets, and if that possibility is not allowed in the model, then a feasible or a satisfactory solution may not exist. Thus, in our model, a penalty is

23

assigned to each voxel based on the dose it receives for a given set of beamlet intensities. Let Fjs denote a convex penalty function for voxel j in structure s of the follwing form: Fjs (zjs ) =

p p 1 ws (Ts − zjs )+ s + ws (zjs − Ts )+ s , vs

where Ts is the dose threshold value for structure s, ws and ps are weighting factors for underdosing, and ws and ps are weighting factors for overdosing. The expression (·)+ denotes max{0, ·}. The function is normalized over the number of voxels in the structure using the coefficient 1/vs . By setting ws , ws ≥ 0 and ps , ps ≥ 1, convexity is ensured. A basic formulation of the FMO problem is then: minimize

S X vs X

Fjs (zjs )

s=1 j=1

subject to zjs =

k X X

Dijs xi

j = 1, . . . , vs , s = 1, . . . , S

h=1 i∈Bθh

xi ≥ 0

i ∈ Bθh , h = 1, . . . , k

The FMO problem is the black-box function F (θ) in the BOO model to quantify the quality of beam vector θ. In contrast with the methods presented by all of the previously cited FMO studies except for Das and Marks [27], Haas et al. [28] and Schreibmann [29], this measure of beam vector quality is an exact measure of the FMO problem, rather than using heuristic methods or scoring approaches which cannot accurately optimize the beam orientations. 2.4

Spatial Considerations

With IMRT optimization, it is possible to generate treatment plans with similar FMO objective function values but very different levels of clinical treatment quality. Chao et al. 2003 [30] illustrate this possibility with two treatment plans that have nearly identical target coverage when plotted on a dose-volume histogram, but while one plan delivers an acceptable homogeneous dose, the other plan results in significant underdosing of the target structure.

24

Chao et al. 2003 [30] show that the probability of microscopic tumor extension decreases linearly with distance from the gross tumor volume, implying that cold spots located near the gross tumor volume are far more likely to allow for tumor metastasis after treatment. Likewise, cold spots located far from the gross tumor volume are unlikely to result in tumor metastasis. To reduce the likelihood of obtaining an unsatisfactory plan with a good dose-volume histograms, spatial coefficients are introduced into the FMO model. For each voxel, we consider its position relative to the primary target as a measure of how acceptable/unacceptable overdosing or underdosing may be. Voxels further from the gross tumor volume are penalized more heavily than voxels closer to the gross tumor because it is less acceptable for a voxel far away from the actual tumor to receive an overdose, as the cancerous cells are unlikely to spread very far from the tumor location (Chao et al. [30]). This additional penalization is called the spatial coefficient, and is denoted cjs for voxel j in structure s. For voxels inside the target structures, the probability of cancer spread is 1, as cancer already exists in those voxels. Let S 0 denote the set of gross tumor structures. Let d`js be the minimum distance from voxel j in structure s to structure `. The spatial coefficient cjs for voxel j in structure s is    1 j = 1, . . . , vs , s ∈ / S0 cjs = n n oo P 0|   min 1, max 0.001, |S [exp (−λ d ) + µ d + β ] j = 1, . . . , vs , s ∈ S 0 , ` `js ` `js ` `=1 where λ` , µ` and β` are weighting coefficients. The objective function for the FMO problem becomes F¯spatial (x) =

S X vs X

cjs Fjs (zjs )

s=1 j=1

2.5

A Primal-Dual Interior Point Algorithm for FMO

To solve the FMO and fractionated FMO models, a primal-dual interior point method is employed. For a convex problem such as the FMO model presented in the preceding section, this method yields an optimal solution in short amount of time.

25

The primal-dual interior point algorithm moves through the interior of the solution space along a central path (a path through the interior of the solution space) toward the optimal solution. The central path is defined by perturbing the KKT conditions described below. These conditions ensure primal feasibility, dual feasibility and complementary slackness. If these conditions are satisfied for a convex programming problem with linearly independent constraints, they yield the optimal solution. Thus, we only need to solve this system to obtain an optimal solution to our FMO model (which has a convex objective function and linear, linearly independent constraints). The KKT system can be difficult to solve, so the conditions are perturbed in order to obtain a solution. The general idea of the primal-dual interior point algorithm is to start from an initial feasible solution, use the perturbed KKT conditions to obtain a step direction close to the central path, and then move the current solution some step length along that direction. The amount of pertubation in the KKT conditions is gradually decreased so that in each step, the solution becomes closer to the optimum. The interior point method allows for the duality gap, the gap between the objective functions of the primal and dual problems, to be calculated, thus providing a measure of how close the current solution is to the optimum. For a problem with continuous variables, when the objective functions of the primal and dual problems are equal (duality gap of zero), the solution is optimal. A mathematical description of the primal-dual interior point method can be found in Nocedal and Wright [31]. Further explanation is provided only as needed to define variables in the algorithm. In the FMO problem, G(x) = −Ix, so the KKT conditions for the FMO formulation are X 1 X Dij Fj0 v s s∈S j∈V s

! X

D`j x`

− si = 0

i ∈ N.

(2–1)

si x i = 0

i ∈ N.

(2–2)

si ≥ 0

i∈N

(2–3)

xi ≥ 0

i ∈ N,

(2–4)

`∈N

26

where the Equation (2–4) ensures that the solution is feasible, as the only constraints in the FMO problem are nonnegativity. The complimentary slackness constraint (2–2) forces the solution to the above conditions to be on the boundary of the solution space. Since a point in the interior of the solution space is desired, the complimentary slackness constraint must be relaxed. The complimentary slackness constraint (2–2) is relaxed by changing each si xi = 0 to si xi = µ, where µ > 0. This, along with requiring that x > 0 and s > 0 for feasibility, ensures that a solution to the perturbed KKT conditions is an interior point. Let n be the size of decision variable vector x. A solution is “close enough” to the central path if the duality measure µ in iteration k is µk =

(xk )> s n

(2–5)

and ||Xk Sk − µk e|| ≤ θµk , where Xk is a matrix with xki values as diagonals and zeros elsewhere, and Sk is a matrix with ski values as diagonals and zeros elsewhere. As the algorithm progresses, µ is reduced to zero until the solution is sufficiently close to optimality. To reduce µ, in each iteration we set µ = µσ, where σ ∈ [0, 1] is called the centering parameter. If the duality gap is very large, σ can be reduced so that µ is reduced faster. In each iteration, the current solution (x, s) is moved in a direction (∆x, ∆s) for some step length α is given by 

 k+1







k

 k

 x   x   ∆x   = +α  sk+1 sk ∆sk Let Xk = diag(xk ), Sk = diag(sk ), H(xk ) = ∇2 φ(xk ). The directions ∆xk and ∆sk can be determined by solving the following equations: h

Xk

−1

i −1 Sk + H(xk ) ∆xk = −rDF − Xk rxs −1 ∆sk = − Xk rxs + Sk ∆xk

27

(2–6) (2–7)

In order to solve this system, we must obtain ∆xk from Equation (2–6) by taking the inverse of [(Xk )−1 Sk + H]. Because computing the inverse of such a large dense matrix is very time consuming, a Cholesky factorization to solve this system quickly. The primal-dual interior point method requires a feasible (x, s) solution in each step. Thus, a maximum step length αmax must be imposed on each step direction to ensure that x ≥ 0 and s ≥ 0: αmax = min

min {−xi /∆xi } , min {−si /∆si }

i=1,...,n

i=1,...,n

Because the inverse of each xi is required to determine the step directions, it is undesirable to have any xi = 0, which would result from using step length αmax . Instead, only a percentage η < 1 of αmax is used: α = min{1, ηαmax }

(2–8)

The benefit of this primal-dual method is that in each step, we can calculate the objective of the dual problem (simply s> x), thus providing a bound on how far the current solution is from optimality. 2.5.1

Primal-Dual Interior Point Algorithm

The primal-dual interior point algorithm is as follows: • Initialization 1. 2. 3. 4. 5.

Select initial values for , σ and η (we use = 5, σ = 0.01, and η = 0.95). Set x0 = 0.05 (very close to 0) and calculate ∇φ(x0 ) and H(x0 ) = ∇2 φ(x0 ). 0 −1 Set s0 = µ(X Pn ) . Set µ0 = ( i=1 ∇φ(x0 )i )/100. Set k = 0.

• Algorithm 1. If the duality gap is very large ((xk+1 )> sk+1 > 107 ), set σ = 0.01σ. 2. Set µk = σµk .

28

3. Solve for the step direction (∆xk , ∆sk ) as described in Equations (2–6) and (2–7). Note that this involves calculating the Hessian H(xk ). 4. Solve for the step length α as described in Equation (2–8). 5. Set xk+1 = xk + α∆x and sk+1 = sk + α∆s. 6. If the duality gap (xk+1 )> sk+1 < , stop. Otherwise, set µk+1 = (xk+1 )> sk+1 /n and k ← k + 1 and repeat. 2.5.2

Hessian Approximations

The most time-consuming step in the primal-dual interior point algorithm is P calculating the Hessian of the objective function in each iteration. For clarity, let P P P denote s∈S 1/vs j∈Vs and Fj00 (x) denote Fj00 ( l∈N Dlj xl ). The Hessian of the FMO problem is then given by 

P

  H(x) =    P

2 Fj00 (x)D1j

... .. .

.. .

Fj00 (x)Dnj D1j . . .

P

Fj00 (x)D1j Dnj .. .

P

2 Fj00 (x)Dnj

     

Note that only the pairwise Dij products differ in each element of the Hessian. By P P P precomputing these cross products, only s∈S 1/vs j∈Vs Fj00 ( l∈N Dlj xl ) has to be recomputed in each iteration. The matrix of the Dij products yields the sparsity (or density) pattern of the Hessian, which stays constant throughout the algorithm. Because the Hessian is symmetric, the matrix values only need to be computed for half of the matrix, further improving efficiency. Despite these observations, computing the Hessian is still so expensive that it renders the algorithm impractical. Methods of approximating the Hessian are implemented to speed up the algorithm. 2.5.2.1

Single Hessian Approximation

One way of speeding up the algorithm is to compute the Hessian just once during initialization to obtain H(x0 ), and then rather than re-compute the Hessian in each iteration, use H(x0 ) as an approximation to H(xk ). We call this the Single Hessian 29

approximation. Although the convergence of such an approximation has not yet been mathematically proven, tests run on several head-and-neck cases for 5-beam and 7-beam plans show that the Single Hessian does in fact converge to the known optimal solution. 2.5.2.2

BFGS Hessian Update

Another Hessian approximation is the Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update. The approximation to the Hessian in iteration k is Bk , with B0 = H(x0 ). The update to the approximated Hessian in each iteration is Bk+1 = Bk +

Bk pk p> qk q> k Bk k − , q> p> k pk k Bk pk

where pk = xk+1 − xk qk = ∇φ(xk+1 ) − ∇φ(xk ) Note that this update ensures that Bk is always symmetric and positive definite, so the Cholesky factorization can still be applied to obtain the step direction. This approximation also empirically converges to the known optimal solution for 5- and 7-beam head-and-neck cases. 2.5.3

Insignificant Beamlets

Insignificant beamlets are those that bear little contribution to the quality of the FMO plan. Letting d denote the diagonal elements of the initial Hessian H(x0 ), the set of insignificant beamlets BI is defined as BI =

|di | i: < 0.001 max{|d|}

These beamlets are removed by removing the ith row and the ith column in H(x0 ) for every i ∈ BI , and then updating the number of bixels to the number of remaining bixels. The insignificant beamlets must be re-inserted into the solution xk in order to calculate

30

the voxel doses, objective function, gradient and Hessian, but the inversion of the Hessian is done to the Hessian with the bad beamlets removed, providing significant time savings. 2.5.4

Warm Start

For the sake of theoretical accuracy, a truly optimal solution cannot have the bad beamlets described in Section 2.5.3 removed. Without removing the bad beamlets a priori, the interior point method must be run for an impractical amount of time to obtain a near-optimal solution, say, = 0.001. The interior point method is typically started with a decision variable vector x equal to almost zero. If the algorithm were to be started at a point closer to the final solution, denoted xwarm , time savings could be gained, allowing all beamlets to be considered in the interior point algorithm in a reasonable amount of time. Such an approach is a called a “warm start”. One difficulty in using a warm start with the interior point method is that a warm start solution may have some xwarm = 0, which is not allowed because the inverse of each i xi must be taken. To correct this problem, any xwarm = 0 is simply replaced with some i very small value γ. Because these zero-valued variables are less important to the problem than nonzero variables, γ should be less than the minimum nonzero value of xwarm . Let > 0}. Then, γ = min{0.001, γ¯ }. : xwarm γ¯ = mini=1,...,n {xwarm i i    xwarm i ∈ / BI i 0 xi =   γ i ∈ BI An additional problem with warm starts in the interior point method is that the KKT variable vector s is unknown at the warm start point. Depending on the algorithm used to obtain the warm start, some information about swarm and µwarm , s and µ at the warm start point, respectively, may not be available. If no information is available about s from the warm start, then s0 = 0. If an interior point algorithm is used to obtain the warm start, then swarm is available. If the warm start did not include the insignificant beamlets, some corrections must be made to account for the insignificant beamlets which will be

31

optimized in the final solution. Let s0 be the initial s used in the interior point method after the warm start has been obtained. Then,    swarm i∈ / BI i 0 si =   µwarm /γ i ∈ BI , where the value chosen for s0i corresponding to insignficant beamlets arises from the general initialization s = µ(X0 )−1 . 2.6

Results

The true Hessian, Single Hessian approximation, and BFGS update implementations of the primal-dual interior point algorithm are tested on six cases head-and-neck cases to obtain coplanar, equi-spaced 5-beam plans. The tests are run on a 2.33GHz Intel Core 2 Duo processor with 2GB of RAM. The method is tested for both leaving in and removing the insignificant beamlets, as well as the proposed alternative to computing the Hessian. The optimality of the interior point method solutions is verified by comparison to the known optimal solutions obtained by Java with CPLEX (ILOG). An acceptable duality gap must be determined in order to implement the interior point method. While we consider a duality gap of = 0.001 to be acceptably close to optimal, it may be unnecessary to achieve such a small duality gap to obtain a quality solution. A duality gap of 0.001 may be sufficiently small to ensure optimal solutions given objective function values using certain weighting parameters, depending on the parameters used in the FMO objective function, the value of the objective function may vary widely. Because of the potential range of values, a stopping criteria based on a relative duality gap rather than an absolute duality gap is preferable. Say the objective function value in an iteration is f . Define the relative duality gap in an iteration to be 0 = /f . An examination of the relative duality gap necessary is presented in Section 2.6.1. Computational results are presented in Section 2.6.2 and clinical comparisons are provided in Section 2.6.3.

32

2.6.1

How Small of a Duality Gap is Necessary?

Because the run time of the algorithm is dependent on the required duality gap, it is desirable to only require the algorithm to achieve as small a duality gap as necessary to ensure a clincally good solution. The duality gap decreases quickly in the first few iterations, and then subsequently decreases by only a small amount per iteration, as shown in Figure 2-1A. If these iterations with only marginal improvements are found to be unnecessary in terms of clinical quality, significant time can be saved by stopping the algorithm once the duality gap is reasonably small, as opposed to waiting until the duality gap is very small. To check the importance of the duality gap, the FMO value and dose delivered to the targets and the saliva glands were plotted against the duality gap in each iteration using the true Hessian and without removing insignificant beamlets. For a representative case, the FMO values per duality gap are shown in Figure 2-1B. It is clear that the duality gap decreases rapidly in the first few iterations, but subsequent iterations yield increasingly smaller drops in the duality gap. Similarly, the amount of dose received by the targets and critical structures does not change significantly toward the end of the algorithm. Figure 2-2 plots the dose received by the two targets, PTV1 and PTV2, starting from a duality gap of 0.15%. The prescription doses are 70 Gy for PTV1 and 50 Gy for PTV2, common dose values used in the cancer clinic at Shands Hospital at the University of Florida. Neither the dose received by 95% of the targets nor the size of the hotspots and coldspots changes significantly in this duality gap range (Figure 2-2A). The hotspots are measured by the percent of the target receiving 110% and 120% of the prescription dose, while the coldspots are measured by the percent of the target receiving at least 93% of the prescription dose (Figure 2-2B). Figure 2-3 shows for two representative cases the amount of dose received by the saliva glands starting from a relative duality gap of 0.15%. Both cases show that the

33

3

0.15

2

0.1

1

0.05

0 0

5

10 15 iterations

20

relative duality gap

FMO value

Objective function and relative duality gap v. iteration 4 x 10 0.3 5 0.25 4 0.2

0 25

Figure 2-1. The duality gap drops sharply in early iterations, but very slowly thereafter. The relative duality gap monotonically decreases after several iterations. change in dose received by the saliva glands as the duality gap decreases is not clinically relevant. From these figures, it appears that a duality gap as large as 0.1% could provide clinically acceptable plans. Since the algorithm may terminate with a duality gap less than the one specified as the stopping criteria, a duality gap larger than 0.1% will also be tested for acceptability. 2.6.2

Computational Results

Table 4-1 shows the average run times for each of the implementations of the algorithm. Relative duality gaps of 0.15%, 0.10%, 0.05% and 0.01%. are compared. The value of θ used to define the central path is 0.5. As expected, using the Single Approximation Hessian alternative with the insignificant beamlets removed is the fastest method, while using the true Hessian is the slowest method, regardless of whether the insignificant beamlets are removed. Interestingly, for large duality gaps, it is slightly faster to leave the insignificant beamlets in the model when using the true Hessian. Otherwise, it is faster to remove the insignificant beamlets. The final FMO values are displayed for each of the tested methods using a duality gap of 0.001, which is sufficiently small to ensure optimal solutions given typical objective function values (Table 2-2). For each case, the final FMO value is nearly identical,

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Target hotspots and coldspots 100

PTV1 PTV2

dose (Gy)

70 65 60 55 50

Percent of target

Target coverage at 95% 75

80 60 40 20 0

0.1 0.05 0 relative duality gap (%)

PTV1 at 1.10 PTV1 at 1.20 PTV1 at 0.93 PTV2 at 1.10 PTV2 at 1.20 PTV2 at 0.93


A B Figure 2-2. Dose received by targets as a function of the duality gap. A) The amount of dose received by at least 95% of each target is used to assess proper target coverage. B) The percent of each target receiving 110% and 120% of the prescription dose indicates hotspots, while 93% of the prescription dose indicates coldspots.

Saliva gland dose at 50% L. parotid gland R. parotid gland L. SMB gland R. SMB gland

25 dose (Gy)

Saliva gland dose at 50%

20 15 10

32

R. parotid gland L. parotid gland R. SMB gland L. SMB gland

30 dose (Gy)

30

28 26 24 22



Figure 2-3. The amount of dose received by at least 50% of each saliva gland remains relatively constant even for large duality gaps. Two representative cases are shown.

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Table 2-1. Average run times for 5-beam treatment plans. Remove insig. Average run time (s) 0 Hessian type beamlets? = 0.001 = 0.15 0 = 0.1 0 = 0.05 0 True no 113.8 55.48 55.48 58.58 True yes 105.6 55.25 56.29 59.09 BFGS no 43.9 13.59 14.17 14.66 BFGS yes 40.9 13.19 13.66 14.30 Single Approx. no 18.1 8.83 8.98 9.29 Single Approx. yes 16.8 8.69 8.84 9.14

= 0.01 71.75 70.56 16.67 15.88 10.13 9.90

Table 2-2. FMO value from using = 0.001.

Hessian type True Hessian True Hessian BFGS update BFGS update Single Approx. Single Approx.

Remove insig. beamlets? no yes no yes no yes

Case 1 2546.22 2546.22 2546.23 2546.24 2546.38 2546.38

Case 2 2200.70 2200.70 2200.70 2200.70 2201.11 2201.15

Case 3 2289.95 2289.95 2289.95 2289.95 2290.40 2290.44

Case 4 2566.38 2566.38 2566.39 2566.39 2566.56 2566.62

Case 5 5024.97 5024.97 5024.97 5024.97 5025.06 5025.14

Case 6 2585.40 2585.40 2585.40 2585.40 2585.82 2585.82

indicating that the Hessian alternatives and the removal of the insignificant beamlets still provide for convergence to the optimal solution. The percentage increases in the FMO values using an absolute duality gap of 0.001 and relative duality gaps of 0.15%, 0.10%, 0.05% and 0.01% are shown in Table 2-3. 2.6.3

Clinical Results

For each of the duality gaps tested, the DVHs of the solutions obtained using the Single Approximation Hessian with the insignificant beamlets removed are compared. Since the each of the interior point implementations obtains nearly identical solutions, it does not matter which implementation is used to produce the DVHs. As previously stated, the prescription doses used are 70 Gy for PTV1 and 50 Gy for PTV2, marked by a vertical line in Figure 2-4A. As saliva glands are the most difficult organs to spare in head-and-neck cases, the only critical structures shown are the saliva glands (Figure 2-4B). All other glands are spared in every implementation. The sparing criteria used for saliva glands is that no more than 50% percent of the saliva gland can

36

Table 2-3. Percent increase in objective function value from various relative duality gaps as opposed to an absolute duality gap of = 0.001. Remove insig. Avg. increase in obj. fn. (%) 0 Hessian type beamlets? = 0.15 0 = 0.1 0 = 0.05 0 = 0.01 True no 0.58 0.58 0.27 0.05 True yes 0.58 0.48 0.25 0.06 BFGS no 0.99 0.54 0.30 0.05 BFGS yes 0.94 0.57 0.26 0.07 Single Approx. no 1.26 0.89 0.60 0.19 Single Approx. yes 1.21 0.87 0.57 0.16 Interior point method: Target DVHs ε’=0.15% ε’=0.10% ε’=0.05% ε’=0.01%

80

100 Volume [Fractional]


Interior point method: Saliva DVHs

60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

80

ε=0.15% ε=0.10% ε=0.05% ε=0.01%

60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

A B 0 Figure 2-4. Quality of DVHs for duality gaps =0.01%, 0.05%, 0.1% and 0.15%. A) The target coverage is nearly identical. B) The saliva gland sparing for the different duality gaps is similar, but the solution for 0 =0.15% sacrifices one saliva gland. The sparing criteria is marked by a star. receive more than 30 Gy in order to be spared. This point is marked by a star in Figure 2-4B. Each of the duality gaps achieves good target coverage. While they each provide similar saliva gland dosage, the plan obtained using 0 = 0.15% slightly surpasses the sparing criteria used for saliva glands. 2.6.4

Spatial Coefficient Results

To assess the possible treatment plan improvement afforded by spatial coefficients, spatial parameters were tuned and then compared to treatment plans obtained without using spatial information. To demonstrate the spatial coefficients, Figure 2-5 displays the

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60 60

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Figure 2-5. The spatial coefficients used for two cases. coefficients used for two cases. In addition to tuning λ, µ and β to values of 1.07, -0.32 and 0.77, respectively, a minimum spatial coefficient of 0.025 was also set for target voxels. By definition, the maximum value of a spatial coefficient is 1. These spatial parameters generally produce treatment plans of nearly identical quality to the best plans obtained without using spatial information, though with the added benefit of preventing misleading dose-volume histograms. In some cases, the spatial coefficients were able to outperform the non-spatial plans. Figures 2-6 and 2-7 illustrates two such cases. In Figure 2-6, the spatial coefficients yield improved target coverage and spare all saliva glands, as opposed to the non-spatial plan which only spares three of the four saliva glands. There is less dose outside the desired target in the plan using spatial coefficients. In Figure 2-7, the spatial coefficients reduce the amount of overdose in the primary targets. In this patient, both the spatial and non-spatial plans spare all saliva glands. 2.6.5

Warm Start Results

Warm start solutions were obtained using the interior point method and the projected gradient algorithm (Nocedal and Wright [31]). The interior point method warm starts were tested with each Hessian possibility and a large duality gap of 200, both with and without insignificant beamlets removed. The projected gradient algorithm was tested using

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Target DVHs: Non−spatial PTV2 PTV1

80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

Saliva gland DVHs: Non−spatial

80

Saliva gland DVHs: Spatial

left parotid gland left submandibular gland right parotid gland right submandibular gland


Volume [Fractional]

100

PTV2 PTV1



Target DVHs: Spatial

60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

80


60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

A B Figure 2-6. Comparison of spatial and non-spatial treatment plans. A) Non-spatial parameters result in slightly low target dosage and fail to spare one saliva gland. B) Spatial parameters allow for improved target coverage and spare all saliva glands.

39

Target DVHs: Non−spatial PTV2 PTV1

80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

80 60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

Saliva gland DVHs: Non−spatial

80

Saliva gland DVHs: Spatial



Volume [Fractional]

100

PTV2 PTV1



Target DVHs: Spatial

60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

80


60 40 20 0 0 10 20 30 40 50 60 70 80 90 Dose [Gy]

A B Figure 2-7. A) Non-spatial parameters result in slightly low target dosage and fail to spare one saliva gland. B) Spatial parameters allow for improved target coverage and spare all saliva glands.

40

several stopping criteria and without insignificant beamlets removed. It was observed that the projected gradient algorithm is fast enough that the time required to remove and re-insert the insignificant beamlets as necessary caused the algorithm to slow down. To be theoretically close to optimal, the interior point method used after the warm start has duality gap of 0.001 and no beamlets removed. The determine the how close the warm start solution is to the final solution, the percent improvement in objective function value the final solution obtains over the warm start is measured. To assess how close to optimality the final solutions using a warm start are, the percentage by which their objective function values are greater than the objective function value of a near-optimal solution is measured. Lastly, the decrease in run times over obtaining a near-optimal solutions are provided. These results for the interior point and projected gradient warm starts are displayed in Tables 2-4 and 2-5, respectively. From Table 2-4, it is clear that using an interior point warm start can provide significant time savings over the near-optimal solution times. There is also a significant increase in the FMO objective function value. From the amount of increase in the objective function value, the interior point warm start does not appear to converge to the optimal solution, and is unlikely to provide acceptable solutions. It is interesting to note that the improvement from the warm start solution to the final solution is very small. This indicates that KKT information obtained from the warm start and used in the final algorithm were unhelpful in improving the solution. For the projected gradient algorithm, once there is less than δ percent decreases from one iteration to the next, the algorithm terminates. Several δ values are tested. As with the interior point warm starts, the projected gradient warm starts also provided significant time savings, as shown in Table 2-5. The final solutions from the projected gradient warm start methods are nearly identical to the near-optimal solutions. The final interior point method also significantly improves the objective value of the warm start solution. This implies that despite not having KKT information about the warm start, the interior point

41

algorithm is still able to converge to the optimal, or at a least near-optimal, solution using the KKT value approximations and adjustments to the warm start vector described in Section 2.5.4.

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43

Table 2-4. Performance measures of interior point method warm starts. interior point warm start final interior point algorithm Improvement Increase Remove insig. Remove insig. over warm start in final 0 0 Hessian type beamlets? Hessian type beamlets? obj. fn. (%) obj. fn. (%) True no 5 true no 0.01 0.00 4.46 True yes 5 true no 0.01 0.19 4.48 True no 5 BFGS no 0.01 0.00 4.79 True yes 5 BFGS no 0.01 0.20 4.84 True no 5 Single Approx. no 0.01 0.00 5.06 True yes 5 Single Approx. no 0.01 0.76 4.49 BFGS no 5 true no 0.01 0.00 4.46 BFGS yes 5 true no 0.01 0.19 4.48 BFGS no 5 BFGS no 0.01 0.00 4.79 BFGS yes 5 BFGS no 0.01 0.20 4.84 BFGS no 5 Single Approx. no 0.01 0.00 5.06 BFGS yes 5 Single Approx. no 0.01 0.76 4.49 Single Approx. no 5 true no 0.01 0.00 4.46 Single Approx. yes 5 true no 0.01 0.19 4.48 Single Approx. no 5 BFGS no 0.01 0.00 4.79 Single Approx. yes 5 BFGS no 0.01 0.20 4.84 Single Approx. no 5 Single Approx. no 0.01 0.00 5.06 Single Approx. yes 5 Single Approx. no 0.01 0.76 4.49

Avg. time savings (s) 64.75 65.20 27.94 28.47 6.85 6.93 64.97 65.09 27.83 28.55 6.90 6.87 64.99 65.00 27.95 28.54 6.88 6.88

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Table 2-5. Performance measures interior point warm start Remove insig. beamlets? δ no 1 no 5 no 10 no 100 no 500 no 1 no 5 no 10 no 100 no 500 no 1 no 5 no 10 no 100 no 500

of projected gradient method warm starts. final interior point algorithm Improvement Increase Remove insig. over warm start in final Hessian type beamlets? 0 obj. fn. (%) obj. fn. (%) True no 0.01 19.83 0.00 True no 0.01 31.78 0.00 True no 0.01 36.43 0.00 True no 0.01 56.59 0.01 True no 0.01 89.46 0.09 BFGS no 0.01 19.83 0.00 BFGS no 0.01 31.78 0.00 BFGS no 0.01 36.43 0.00 BFGS no 0.01 56.59 0.03 BFGS no 0.01 89.46 0.13 Single Approx. no 0.01 19.82 0.00 Single Approx. no 0.01 31.77 0.01 Single Approx. no 0.01 36.42 0.01 Single Approx. no 0.01 56.56 0.08 Single Approx. no 0.01 89.44 0.27

Avg. time savings (s) 36.63 10.98 19.16 39.28 56.88 9.27 12.53 19.30 27.79 30.30 3.40 3.95 4.28 9.28 10.04

2.7

Conclusions

The primal-dual interior point method is an effective algorithm for obtaining fluence maps that deliver quality treatment plans. The proposed Hessian alternatives appear to converge to the optimal solution, even when insignificant beamlets are removed. The removal of the insignificant beamlets provides significant time savings in all instances. The interior point method may also be run with a duality gap as large as 20 and still achieve quality treatment plans, thus decreasing the amount of time required to run the algorithm. Of the implementations tested, the fastest method that still provides quality solutions without using a warm start is to use the Single Approximation Hessian alternative, remove insignificant beamlets and employ a relative duality gap of 0.1%. When the interior point method is started with one of the warm starts discussed, time savings were again significant. Although the interior point warm starts generally provided more improvement in computation time than the project gradient warm starts, the final solutions using the projected gradient warm starts were much closer to optimality. The fastest and most effective warm start method is to use the projected gradient algorithm with δ = 500, followed by the interior point method with = 0.1% and the Single Approximation Hessian. This combination results in a near-optimal solution with an average total computation time of 8.32 seconds.

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CHAPTER 3 BEAM ORIENTATION OPTIMIZATION 3.1

Introduction

In a typical head-and-neck treatment plan, radiation beams are delivered from 5-9 nominally-spaced coplanar orientations around the patient. These coplanar orientations are obtained from rotating the gantry only. As shown in Figure 3-1, several components of a linear accelerator can rotate and translate to achieve more orientations than those obtained from rotating the gantry. The available orientations consist of the orientations obtained from rotation of the gantry, collimator and couch, as well as the three translation directions of the couch.

Figure 3-1. A linear accelerator and the available movements; the gantry rotation is highlighted. BOO is the problem of selecting from the available beam orientations the best set to use in delivering a treatment plan. Given a fixed set of beams, different fluence maps (radiation intensities of beamlets) yield treatment plans with different qualities. Thus, the quality of an optimized fluence map should be considered when selecting a set of beam orientations to use in a treatment plan.

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3.2

Literature Review

Many approaches have been taken to solve the BOO problem. Evolutionary algorithms (Schreibmann [29]) and variants of evolutionary algorithms, particularly genetic algorithms (Ezzell [32], Haas et al. [28], Li et al. [33]) have been employed. Li et al. [34] use a particle swarm optimization method, which is conceptually based on evolutionary algorithms. Bortfeld and Schlegel [35], Djajaputra et al. [36], Lu et al. [37], Pugachev and Xing [38], Rowbottom et al. [39] and Stein et al. [40] have all employed variations of simulated annealing to determine a beam solution. Söderstrom and Brahme [41] selected coplanar beam orientations using two measures, entropy and the integral of the low frequency part of the Fourier transform of the optimal beam profiles, both of which are based on the size and shape of the target structure. Soderstrom and Brahme [42] also use an iterative technique to determine the optimal number of coplanar beams required using BOO. Das and Marks [27] use a quasi-Newton method. Rowbottom et al. [43] use artificial neural network algorithms to select beam orietations. Gokhale et al. [44] use a measure of each beam’s “path of least resistance” from the patient surface to the target location to determine the best beam directions. Meedt et al. [45] use a fast exhaustive search to obtain a non-coplanar solution. The concept of beam’s-eye view (BEV) has also been commonly used to approach the BOO problem (Chen et al. [46], Cho et al. [47], Goitein et al. [48], Lu et al. [37], Pugachev and Xing [38, 49, 50]). Despite the varying techniques to quantify the quality of a beam solution, it is widely accepted that the optimal solution to the FMO problem presents the most relevant measure (Bortfeld and Schlegel [35], Djajaputra et al. [36], Holder and Salter [51], Lee et al. [20, 21], Li et al. [33, 34], Meedt et al. [45], Morrill et al. [52], Oldham et al. [53], Rowbottom et al. [39, 43, 54], Schreibmann et al. [29], Söderstrom and Brahme [41], Stein et al. [40], Wang et al. [55, 56], Woudstra and Heijman [57]). Given this accepted measure of treatment quality, the shortcoming of the previous works is twofold. First, they predominantly only consider coplanar angles, and not necessarily even the entire

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coplanar solution space, while those that do consider non-coplanar beams only consider a hand-selected subset of the available orientations. Second, the majority of the previous studies do not select beam solutions using the FMO problem as a model for determining quality; instead, the beam solutions are chosen based on scoring methods (e.g., BEV, path of least resistance) or approximations to the FMO. By not optimizing the beam solution with respect to the exact FMO problem, the BOO methods cannot guarantee convergence to an optimal solution. Of the previously cited works, only Das and Marks [27], Gokhale et al. [44], Meedt et al. [45], Lu et al. [37], Rowbottom et al. [39] and Wang et al. [56] consider non-coplanar orientations. This is likely due to the computational difficulties associated with the inclusion of non-coplanar orientations as well as the widespread belief that non-coplanar orientations do not improve the quality of a treatment plan. Also, of those works that addressed non-coplanar beams, Das and Marks [27] require that the beam distances be maximized, essentially requiring that beam solutions must be equi-distant and thus restricting the size of the solution space; Meedt et al. [45] only consider 3,500 beams (a minute subset of orientations available by rotation of the couch and the gantry); and Wang et al. [56] use only nine pre-selected non-coplanar beams. With the exception of Das and Marks [27], Haas et al. [28] and Schreibmann [29], the previous studies have based their BOO approaches not on a beam solution’s optimal solution to the FMO problem, but on locally optimal FMO solutions or on various scoring techniques. Without basing BOO on the optimal FMO solutions, the resulting beam solutions have no guarantee of optimality, or even of local optimality. 3.3

Model Formulation

The goal of radiation therapy treatment planning is to design a treatment plan that delivers a prescribed level of radiation dose to the targets while simultaneously sparing critical structures by ensuring that the level of radiation dose received by these structures is less than a structure-specific radiation dose. These two goals are contradictory if the

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targets are located near critical structures. This is especially problematic for certain cancers, such as tumors in the head-and-neck area, which are often located very close to, for instance, the spinal cord, brain stem and salivary glands. In order to model the BOO problem, a quantitative measure that appropriately makes trade-offs between these contradictory goals must be developed. Let F (θ) be a black-box function that quantifies the quality of the treatment plan if radiation is delivered from beam vector θ = (θ1 , . . . , θk ), where k is the user-specified number of orientations that may be used. F is formulated in such a way that the optimal plan yields the minimum function value. For k beams orientations to be optimized in the treatment plan, the vector of decision variables representing the beam orientations is defined as θ = (θ1 , . . . , θk )T . The decision vector θ is used as input into the black-box function F (θ) to determine the ability of the beam vector to deliver the prescribed treatment without unduly damaging normal tissue and critical structures. The BOO problem is then formulated as min F (θ) subject to θh ∈ B

h = 1, . . . , k,

where B is the set of candidate beams. The candidate set of beams can be selected according to any user-specified criteria; for example, the beams can be coplanar or non-coplanar, continuous or discrete, or only represent a subset of the available beams. It is also possible to fix some beams and only optimize a subset of the total number of beams to be used. Theoretically, the linear accelerator is able to capture a continuous set of orientations, but due to machine tolerances, the actual beams delivered may not be exactly the desired beams. Therefore, it is common to only consider a discretized set of beam orientations. In our BOO model, the black-box function F (θ) is the convex FMO problem described in Section 2.3, thus ensuring an exact measure of the quality of each beam vector. Even though F (θ) is convex, this formulation of the BOO problem is fundamentally

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nonlinear because the physics of dose deposition change with each beam orientation; that is, the effect of a beam on each patient can be drastically different than the effect of a neighboring beam. To illustrate the nonlinearity of the problem, Figure 3-2 shows the FMO problem as a function of just two coplanar beam angles. From this illustration, it is evident that the FMO function, particularly in higher, more realistic dimensions, is likely to also be multi-modal. Although the FMO problem itself can be solved quickly using the convex model presented in Section 2.3, in order to perform the FMO, lengthy calculations must be made in order to determine each candidate beam’s effect on the patient. These calculations, described in Section 3.5, require ≈ 13 minutes per beam to calculate, and thus make each evaluation of the FMO problem expensive. Despite the time required for each function evaluation, the limiting factor in beam orientation optimization is the hard drive space required to store the beam data for each candidate beam. If the candidate set of beams is small, this data can be pre-computed and stored, allowing the FMO problem to be solved quickly in the BOO problem. But, if the candidate set of beams is large—for example, consisting of non-coplanar orientations—then the data cannot be pre-computed due to storage requirements. Because of these difficulties with the BOO problem, previous studies have been largely unable to consider the entire solution space of available beams. By using the response method, which is specifically designed to model expensive nonlinear black-box functions, we can iteratively identify promising beam vector solutions and generate beam data for these solutions on-the-fly, thus circumventing the issue of storage space and allowing for the consideration of all deliverable beam orientations. 3.4

Mixed-Integer Model Formulation

As an alternative to the BOO model given in Section 3.3, if the set of beam orientations B is finite, the BOO and FMO problems can be formulated together and solved simultaneously as a mixed-integer linear or nonlinear program (D’Souza et al. [58],

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Figure 3-2. FMO value as a function of two angles. Ehrgott and Johnston [59], Ferris et al. [17], Lee et al. [20, 21], Lim et al. [60], Shepard et al. [22], Wang et al. [61]). The FMO formulation can be combined with BOO in the following model. Let yθ be a binary variable indicating whether or not beam θ ∈ B is used. If beam θ is used in the treatment plan, then all the beamlets in θ, Bθ , are “turned on”; that is, they can have positive fluences up to some pre-determined maximum intensity M . The simultaneous BOO+FMO MIP model is then minimize F (z) subject to zjs =

k X X

Dijs xi

j = 1, . . . , vs , s = 1, . . . , S

h=1 i∈Bθk

x i ≤ M yθ X

i ∈ Bθ , θ ∈ B

yθ ≤ k

θ∈B

xi ≥ 0

i ∈ Bθ , θ ∈ B

yθ ∈ {0, 1}

θ∈B

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In order to solve such a problem, all beam data must be pre-computed for every beam orientation. As described in Section 3.5, beam data requires a tremendous amount of time and space to compute and store. Because of this requirement, only a small subset of all possible beam orientations can be considered due to time and space constraints for a BOO+FMO MIP formulation. 3.5

Beam Data Generation

For each beam orientation that is considered, lengthy calculations must be made to determine the beam’s effect on the patient’s tissue and organs. This includes determining in which structure each voxel lies, which voxels are hit by which beamlets and the amount of intensity of each beamlet is deposited in each voxel through which it passes. Beamlet dose computation models used in IMRT rely heavily on ray-tracing algorithms for voxel classification and determination of the radiological path (Fox et al. [62]). Voxel classification (Siddon [63]) establishes whether voxels are inside or outside the path of a radiation beam and classifies voxel centers as inside or outside of segmented targets and critical structures. The radiological path is the effective distance traveled by a beamlet when the effect of traveling through tissues of different densities is considered. The exact radiological path of a beamlet through the patient is required to correct for tissue heterogeneities in determining the dose deposition coefficients (Siddon [64]). Siddon’s ray-tracing algorithms (Siddon [63, 64]) have been the standard methods used for ray-tracing in radiotherapy since the 1980s. In Siddon’s polygon and voxel ray-tracing algorithms for voxel classification (point-in-polygon testing), structures are represented as 3D polygonal objects, known as Siddon Prisms, and the signs of cross-products of rays passing through the polygons are used to determine whether a voxel lies inside or outside a structure. Despite its overwhelming use, Siddon’s algorithm for polygon ray-tracing becomes very costly due to the number of voxels in a patient. Fox et al. [62] developed a novel approach to polygon ray-tracing that circumvents the need for cross-products by translating the polygon structure onto a coordinate system, replacing

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the need for a cross-product by the sign of the second coordinate of each voxel in the coordinate system. In Siddon’s algorithm for determining radiological paths (Siddon [64]), the radiological path must be determined for each voxel for every beamlet. This involves computations for millions of beamlet-voxel combinations. As reported by Jacobs et al. [65] a significant amount of computational time is required for these repeated calculations. Fox et al. [62] combine the incremental voxel ray-tracing algorithm presented by Jacobs et al. [65] with a method of virtual stereographic projection to significantly reduce the computational cost of obtaining radiological path lengths. Using their polygon translation and incremental ray-tracing algorithms, Fox et al. [62] achieve a 100-300 fold improvement in computation time over Siddon’s point-in-polygon algorithm. Because of the significant reduction in computation time, these methods are used to generate beam data. Because these beam data calculations must be performed for each of millions of beamlet-voxel combinations, beam data generation is a lengthy process, requiring ≈ 13 minutes per beam using the algorithms described by Fox et al. [62]. In a typical FMO formulation, the beam vector is pre-determined and the beam data for the beam vector is calculated once and stored a priori. For a 5-beam case, this requires ≈150 MB of space to store. As with a typical FMO problem, in a simultaneous FMO+BOO mixed-integer programming (MIP) formulation, beam data for each of the candidate beams in B must be generated a priori. If candidate beams are considered only for coplanar angles on a 10◦ grid, that is, only every 10th angle, beam data would have to be computed for 36 beams, which requires ≈5 hours to compute and ≈800 MB of space to store. If we also wanted to consider the possibility of rotating the couch on a 10◦ grid in addition to the gantry, beam data would then have to be computed for 362 beams, which would require ≈170 hours and ≈ 60 GB of space for just one plan.

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Clearly, the storage space requirements for each beam restricts the number of beams that can be considered in a simultaneous FMO+BOO MIP formulation. This issue is typically addressed by simply restricting the number of candidate beams in B. Lee et al. [20] restrict the set B to only contain 18 pre-selected beam orientations, which can be coplanar or non-coplanar. If only gantry and couch rotations are allowed on a 10◦ grid, a beam set of 18 beams comprises only a small percent of the available beam orientations. As more ranges of motion are allowed, this percentage falls even further. The inclusion of all possible beam orientations significantly increases the size of the solution space and could possibly allow for improved treatment plans, but the beam data for all orientations cannot be pre-computed. In order to consider these orientations, we use a method that allows us to generate the beam data on-the-fly only as necessary. 3.6

A Response Surface Approach to BOO

The shortcoming of the previous works on BOO is twofold. First, they predominantly only consider coplanar angles, and not necessarily even the entire coplanar solution space, while those that do consider non-coplanar beams only consider a hand-selected subset of the available orientations. Second, the majority of the previous studies do not select beam solutions using the FMO problem as a model for determining quality; instead, the beam solutions are chosen based on scoring methods (e.g., BEV, path of least resistance) or approximations to the FMO. By not optimizing the beam solution with respect to the exact FMO problem, the BOO methods cannot guarantee convergence to an optimal solution. Of the previously cited works, only Das and Marks [27], Gokhale et al. [44], Meedt et al. [45], Lu et al. [37], Rowbottom et al. [39] and Wang et al. [56] consider non-coplanar orientations. Of these works, Das and Marks [27] require that the beam distances be maximized, essentially requiring that beam solutions must be equi-distant and thus restricting the size of the solution space; Meedt et al. [45] only consider 3,500 beams (a

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minute subset of orientations available by rotation of the couch and the gantry); and Wang et al. [56] use only nine pre-selected non-coplanar beams. With the exception of Das and Marks [27], Haas et al. [28] and Schreibmann [29], the previous studies have based their BOO approaches not on a beam solution’s optimal solution to the FMO problem, but on locally optimal FMO solutions or on various scoring techniques. Without basing BOO on the optimal FMO solutions, the resulting beam solutions have no guarantee of optimality, or even of local optimality. Because beam data generation is costly, a method that iteratively identifies only promising beam orientations is required. The response surface (RS) method is such an algorithm. In contrast to the previous studies, our approach to the BOO problem allows for the inclusion of all possible beam orientations which are measured according to the exact FMO problem, thus ensuring convergence to optimality due to the properties of the response surface method. The RS method is designed to efficiently model expensive black-box functions. In this application, the FMO solver is our black box and the set of beams to be used is the input. As in Aleman et al. [66, 67], we employ the response surface method as detailed in Jones [68] and Jones et al. [69]. 3.6.1

Overview of Response Surfaces

The response surface method identifies promising solutions based on the performance of previous solutions. The function value and expected improvement over the current best solution of a certain point is estimated based on the function behavior learned from previously sampled points and their calculated objective function values. The function values of points are related by correlation functions that depend on each point’s distance from the previously sampled points. From the correlation functions, the algorithm predicts the probability that the best solution will improve at unexplored points in the solution space. Using this probability, a promising solution is identified. For the BOO problem,

55

beam data only needs to be generated for these promising solutions, thus saving both computation time and storage space. The response surface method models the objective function as a stochastic process of the form F (θ) = µ + (θ),

(3–1)

where µ is a constant representing an average of the function F and (θ) is a random error term associated with the point θ. In the general case, the error terms between two points, say θ (1) and θ (2) , are correlated by h i Corr θ (1) , θ (2) = exp −d θ (1) , θ (2) ,

(3–2)

where d(θ (1) , θ (2) ) is a weighted distance measure between θ (1) and θ (2) . Intuitively, if two points are very close together, the correlation between them will be close to one; similarly, if two points are very far apart, the correlation between them will approach zero. Jones et al. [69] propose the following weighted distance measure in general:

(1)

d θ ,θ

(2)

=

k X

p (1) (2) h c h θh − θh ,

h=1

where the parameters ch and ph are weighting factors corresponding to the importance of each variable h and the smoothness of the function F in the direction of variable h, respectively. If small changes in variable h cause large changes in the function F , then ch should be large to reflect that two points with relatively small differences in the value of variable h should be “far” apart due to the large difference in their function values, and thus have a low correlation. The parameter ch can take on any value, whereas 1 ≤ ph ≤ 2, with ph = 2 corresponding to objective function smoothness and ph = 1 corresponding to less objective function smoothness. In the application to BOO, θ = (θ1 , . . . , θk ) is the vector of k angles from which radiation will be delivered. Because no beam is more important than another beam, each beam orientation h contributes equally to the FMO function, so ch = c and ph = p for

56

all h = 1, . . . , k. To maintain tractability of the subproblems described in the following sections, the angles are treated as though they are points on a line rather than points on a circle and so a Euclidean distance metric is used to determine the distance between two points. The weighted distance measure for BOO is then

p

d θ (1) , θ (2) = c θ (1) − θ (2) ,

(3–3)

p

where k · kp denotes the `p -norm. To ensure tractability of the subproblems described in Section 3.6.2, the value p = 2 is used. The idea of the RS method is to iteratively evaluate the true function F at certain beam vectors θ, and then construct the conditional stochastic process given these function values. This conditional stochastic process is then used to decide where to evaluate the function F next. Due to the time and space required to generate the beam data necessary to evaluate the function F , it is desirable to only evaluate points that will either improve the best solution with a significant probability or significantly increase our knowledge of the function. The optimization models to determine the next observation are described in Section 3.6.2. Let θ (1) , . . . , θ (n) be n previously sampled points. Rn is the matrix of correlations between the previously sampled points, yn is the vector of function values F (θ (i) ) of the previously sampled points and µ ˆn and σ ˆn be estimators of the average and variance of the function F , respectively. The response surface algorithm is given by: • Initialization: 1. Choose values for the parameters c and p. 2. Choose an initial sample size, n, and a set of angles θ (i) , i = 1, . . . , n. Evaluate the function F at each of these points, yielding the values yi , i = 1, . . . , n. • Iteration: 1. Compute or update the values of Rn , R−1 ˆn , σ ˆn , and F n , the minimum n , µ observed objective function value.

57

2. Determine the next point to observe using one of the methods described in Section 3.6.2 and call this point θ (n+1) . 3. Find the value yn+1 = F (θ (n+1) ), set n ← n + 1, and repeat. 3.6.2

Determining the Next Observation

Because the function F is expensive to evaluate, we want to sample as few points as possible. Thus, in each iteration, an optimization problem is solved that determines the “best” next point at which to observe the true function F . Some of the optimization problems that have been proposed in the literature depend on the uncertainty of the predictor as a function of θ, as well as the expected improvement over the current best solution (Jones [68], Jones et al. [69]). Let rn (θ) be the vector correlations between θ and the n previously sampled points. The uncertainty is then given by " s2n (θ) = σ ˆn2

2 # > −1 r (θ) 1 − 1 R n n , 1 − rn (θ)> R−1 n rn (θ) + > 1 R−1 n 1

where σ ˆn2 =

1 (yn − 1ˆ µn )> R−1 µn ) n (yn − 1ˆ n

is the estimator of the variance σn2 based on the n observations. The expected improvement, denoted In (θ), is given by In (θ) = sn (θ) [zΦ (z) + φ (z)]

(3–4)

where z=

F n − Fˆn (θ) sn (θ)

! (3–5)

and F n = min{y1 , . . . , yn } is the current best solution and Fˆn (θ) is the estimated function value of θ given the n previously sampled points. Φ and φ are the c.d.f. and p.d.f. of a standard normal random variable, respectively.

58

The selection of the next point will be based on selecting the point that maximizes either the uncertainty or the expected improvement, or a combination of both. Denote the beam vector to be chosen as the vector θ. 3.6.2.1

Maximizing the expected improvement

Jones [68] and Jones et al. [69] recommend selecting the next point to sample as the point θ for which the expected improvement over the current best solution value, In (θ), is largest. This corresponds to solving the following optimization problem: max In (θ) subject to θh ∈ B

h = 1, . . . , k

Although this is a difficult optimization problem, it can be solved using a branch-and-bound technique, but in order to do so, an upper bound on In (θ) must be obtained. This can be done by solving for the expected improvement in equation (3–4) while substituting an upper bound on the uncertainty and a lower bound on Fˆn (θ), used in equation (3–5) to determine the value z. The method of bounding Fˆn (θ) is taken directly from Jones [68] and Jones et al. [69] and is not discussed further here. The method of bounding s2n (θ) is improved from the original formulation in Jones et al. [69] to overcome numerical instabilities, and is presented in Section 3.6.2.2. The branch-and-bound algorithm used to maximize In (θ) is described in Section 3.6.2.3. 3.6.2.2

Obtaining an upper bound on the uncertainty

Due to the complexity of the s2n (θ) function, maximizing the uncertainty is a difficult problem to solve. It can be relaxed into a linearly constrained quadratic programming problem as follows (Jones et al. [69]). The resulting solution to the relaxed uncertainty maximization problem is an upper bound on the uncertainty that can be used in determining an upper bound on In (θ) as described in Section 3.6.2.1. Let r = {r1 , . . . , rn }, where r is a vector of decision variables independent of θ. By treating both r and θ as decision variables, a quadratic objective function is obtained.

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Because r is now a decision variable independent of θ, an equality constraint must be added to the problem to ensure that r assumes the correct correlation values according to the correlation definition in equation (3–2). This constraint is nonlinear, but it can be relaxed by expressing the single equality as two inequalities (≤ and ≥) and then replacing the nonlinear terms generated by ln(ri ) and ckθ − θ (i) k22 with linear underestimators ai + bi ri and pi,h + qi,h θh , respectively. The different types of linear estimators require different values for ai , bi , pi,h and qi,h , and are differentiated by a superscript c for the chord underestimators and a superscript t for the tangent line underestimators in the model formulation, denoted Problem s2 -UB. Unfortunately, this relaxation provided by Jones et al. [69] can become numerically unstable if two sampled points are very close together. If such a situation arises, the bounds of the corresponding correlation value can become so close that due to round-off error, the lower bound riL can become slightly larger than the upper bound riU , resulting in infeasibility. To avoid such an instability, instead of bounding ri using constraints, the amount by which ri is outside of its feasible range is penalized by adding penalization terms wiL = min{0, ri − riL } and wiU = min{0, riU − ri }. This final formulation is given in Problem s2 -UB. This formulation has only two more variables and two more constraints for each sampled point, so the increased problem size does not significantly increase the amount of time required to solve the problem.

60

PROBLEM s2 -UB: Choose r and θ to " # n n > −1 2 X X 1 − 1 R r n L 2 U 2 min − σ ˆn2 1 − r> R−1 r + + w + w n i i 1> R−1 n 1 i=1 i=1 subject to

(aci

+

bci ri )

+c

ati + bti ri + c

k X h=1 k X

t pti,h + qi,h θh ≤ 0

i = 1, . . . , n

c θh ≤ 0 pci,h + qi,h

i = 1, . . . , n

wiL ≤ 0

i = 1, . . . , n

wiL ≤ ri − riL

i = 1, . . . , n

wiU ≤ 0

i = 1, . . . , n

wiU ≤ riU − ri

i = 1, . . . , n

lh ≤ θh ≤ uh

h = 1, . . . , k

h=1

Using the upper bound on the uncertainty provided by Problem s2 -UB, the point yielding the maximum uncertainty is obtained by using the same branch-and-bound method described in 3.6.2.3, except that s2n (θ) is maximized rather than In (θ). Alternatively, another approach would be to choose the next point based on maximizing uncertainty rather than the expected improvement. The branch-and-bound approach described in Section 3.6.2.3 can be adapted to solve that problem rather than maximizing the expected improvement. 3.6.2.3

Branch-and-Bound

A branch-and-bound method is used to determine the maximum expected improvement in each iteration. At some point in the algorithm, n points, θ (1) , . . . , θ (n) , have already been observed. The solution space is divided into regions based on these previously sampled points and consider each region as a separate subproblem. Each of these subproblems is solved using branch and bound. First, the upper bound on the uncertainty is determined as described in Section 3.6.2.2 using the subregion’s

61

lower and upper bounds on θ. Next, the lower bound Fˆ L on Fˆn (θ) is determined using the method in Jones [68] and Jones et al. [69]. The upper bound on s2n (θ) and lower bound on Fˆ are now used to determine an upper bound on In (θ) over the current subregion by solving for In (θ) substituting Fˆn (θ) = Fˆ L and sn (θ) = sU as described in Jones [68] and Jones et al. [69]. In addition, the θ that yielded the maximum uncertainty can be used to evaluate the function In (θ), yielding a lower bound on In (θ) over the interval lh ≤ θh ≤ uh , h = 1, . . . , k. This value is used to update the current best lower bound found (i.e., if the current best lower bound is less than the new lower bound found, the current best lower bound is replaced by the new one; otherwise, the current best lower bound is unchanged). If the upper bound is less than the current best lower bound, the subregion is discarded as not interesting. If the lower and upper bound are very close, we say that we have found the optimum over the current subregion. Otherwise, the upper bound is significantly larger than the current lower bound, so the subregion is further divided into subregions as described below and the procedure is repeated for each of the new regions. This is the branching step. At some point, there are no more subregions to consider, as we have either decided they are not interesting or have found the optimal solution for that subregion. Then, the algorithm terminates and the current best lower bound is the optimal solution for In (θ) over the current region. This branch-and-bound procedure is applied to each of the regions, and the overall largest In (θ) value is then the maximum In (θ), and the corresponding θ is the next point at which to evaluate the FMO function. Selecting the subregions. An important component of the branch-and-bound algorithm is the method of selecting the subregions. The definition of these subregions, as well asl the order in which they are explored, can have significant impact on both the amount of time and memory required to perform the algorithm. As our implementation

62

of the branch-and-bound method requires that the entire solution space be divided into subregions before the branch-and-bound algorithm begins, the selection of these initial regions may also affect the speed of the algorithm. Initial regions. Before beginning the branch-and-bound process, the solution space of the decision variables, θh ∈ [0, 360] for all h = 1, . . . , k, is divided into a set of initial regions. If θ represents non-coplanar orientations, we consider two ways of selecting the regions defined by the non-coplanar orientations. First, we consider the entire solution space as the only region, that is, instead of dividing the solution space into several subregions, we only consider one subregion that encompasses the entire solution space (see Figure 3-4A). ¯ ∈ H, Second, denote a subset of variable indices H ⊆ {1, . . . , k}. For each index h ¯ For each previously sampled order the n previously sampled points increasingly by h. point i = 1, . . . , n − 1, consider the regions defined by lh = 0 and uh = 360 for h ∈ / H, (i)

(i+1)

and lh¯ = θh¯ and uh¯ = θh¯

. Also consider the region defined by lh = 0 and uh = 360 (1)

for h ∈ / H, and lh¯ = 0 and uh¯ = θh¯ . Similarly, consider the region defined by lh = 0 and (n)

uh = 360 for h ∈ / H, and lh¯ = θh¯ and uh¯ = 360. Figures 3-4A-3-4D illustrate the initial regions for different H values where k = 2. Denote the initial region set where H = ∅ as B0 (Figure 3-4A), H = {1} as B1 (Figure 3-4B), H = {2} as B2 (Figure 3-4C) and H = {1, 2} as B2 (Figure 3-4D). Note that in the coplanar case, it is only necessary to test the initial region scheme for one angle because the angles are interchangeable. Bounds for discrete and continuous variables. If θ is discrete, the points on the boundary between between the two subregions will be contained in both subregions, thus (1)

creating an inefficiency. This can be seen in Figure ??, where θ b is the point at which we branch and the blue line represents the division of the region into two subregions. The boundary line is contained in both the top interval and the bottom interval. This overlap can be avoided when θ is integral by adjusting the bounds between subregions in such a

63

Initial region scheme B1 360

300

300

240

240 Couch angle

Couch angle


180

180

120

120

60

60

0

0

60

120

180 240 Gantry angle

300

0

360

0

60

120

A

300

300

240

240

180

120

60

60

120


300

360

180

120

60

360


Couch angle

Couch angle

Initial region scheme B2

0

300

B

360

0


300

0

360

C

0

60

120


D

Figure 3-3. Initial regions in the branch-and-bound algorithm. A) Initial regions with H = ∅ (B0). B) Initial regions with H = {1} (B1). C) Initial regions with H = {2} (B2). D) Initial regions with H = {1, 2} (B3).

64

way as to prevent overlapping between any subregions. If the lower bound lh on θh in a subregion is fractional, then we discard the non-integral solutions by setting lh = dlh e. Similarly, if the upper bound uh on θh in a subregion is fractional, then uh = buh c. If the lh and uh bounds are integral and lh = uh , overlapping is avoided by setting lh = lh − 1 (see Figure ??). If θ is continuous, the bounds cannot be adjusted. Branching scheme. The basic principle of the branch-and-bound method is to decompose regions into smaller subregions in such a way that as many subregions as possible can be discarded as uninteresting, leaving a reduced number of subregions that must actually be searched. The branch-and-bound method is a well studied problem, and as such, there are numerous methods of selecting the subregions. Regions may be divided into two equal subregions (bisection), or more generally, into multiple subregions which may or may not be equal in size (multisection) (Csallner et al. [70], Lagouanelle and Soubry [71]). Some other common methods include selecting only a subset of variables on which to branch (Epperly et al. [72]), using Langrangian duality to obtain lower bounds (Barrientos and Correa [73], Thoai [74], Tuy [75]) and applying decomposition algorithms (Phong et al. [76], Bomze [77], Cambini and Sodini [78]). In our branching step, we form the subregions based on some point in the region. The (1)

region is divided at this point along one of the indices. In Figure 3-4A, θ b is the point at which we branch. We branch by dividing the region horizontally into two subregions (1)

at θ b , taking into account the adjustments to the bounds described above so as to avoid overlapping regions. For k = 2, in each branching step, we alternately divide the region horizontally (along index 2) and vertically (along index 1) as shown in Figures 3-4B–3-4D. (1)

After branching horizontally once at θ b as shown in Figure 3-4B, we examine the top (2)

region and select θ b as the point at which we branch. We then branch by dividing this (2)

(3)

subregion vertically at θ b . We proceed in the same manner for θ b , where we branch horizontally, and so on until the convergence criteria is met.

65

In the general case, we divide the region into two subregions along the branching ¯ = 1, . . . , k sequentially. For the index while cycling through each of the indices h ¯ the bounds for one new subregion are lh¯ = lh¯ and uh¯ = θb,h¯ − 1, branching index h, and the bounds for the other new subregion are lh¯ = θb,h¯ and uh¯ = uh¯ . The lower and upper bounds on the region for the remaining indices are unchanged for both new ¯ subregions, i.e. lh = lh and uh = uh for h 6= h. In the non-coplanar case, a beam in θ may be represented by more than one index. For example, if a single non-coplanar beam consisting of couch and gantry rotation is optimized, the vector θ consists of θ1 representing the gantry angle and θ2 representing ¯ ∈ {1, 2} represents branching on either the the couch angle. The branching index h gantry angle or on the couch angle. If two such non-coplanar beams are optimized, then θ consists of θ1 and θ2 representing the gantry and couch angles of the first beam, respectively, and θ3 and θ4 representing the gantry and couch angles of the second beam, ¯ ∈ {1, 2, 3, 4} then represents branching on a single respectively. The branching index h component of a single beam. Accounting for symmetry. In the case where θ represents a set of coplanar beam angles, the ordering of the variables in θ is irrelevent to the FMO value obtained at θ. For example, if θ (1) = (10, 20, 30, 40) and θ (2) = (20, 30, 40, 10), then F (θ (1) ) = F (θ (2) ). Thus, it is redundant to consider both θ (1) and θ (2) , and elimination of these redundant regions can greatly decrease the size of the solution space. For example, if we consider the two-dimensional case (k = 2), the solution space is a square region with 0 ≤ θ1 ≤ 360 and 0 ≤ θ2 ≤ 360. The points above the line θ1 ≤ θ2 are equivalent to the points below the line, so we only need to consider one of these regions. Say we branch by splitting the region into four equal quadrants, as shown in Figure 3-5A. If we arbitrarily choose to only examine the points above the line θ1 ≤ θ2 , then quadrant 4 can be eliminated.

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Branching scheme

Branching scheme

Couch angle

u2

Couch angle

u2

θ(1) b

l2

θ(1) b

l2 l1

u1

Gantry angle

l1

u1

Gantry angle

A

B

Branching scheme

Branching scheme

u2

u2

θ(3) b

Couch angle

θ(2) b

Couch angle

θ(2) b

θ(1) b

l2

θ(1) b

l2 l1

Gantry angle

u1

l1

C

Gantry angle

u1

D

Figure 3-4. Partitioning a region into subregions. A) Partitioning a region into subregions without accounting for overlap. B) Preventing overlapping regions. C) Regions after two branches. D) Regions after three branches.

67

A

B

Figure 3-5. Accounting for symmetry. A) Accounting for symmetry in 2D. B) Accounting for symmetry in 3D. In three dimensions, the solution space is a cube. If we branch by splitting the cube into eight equal cubes, the region to be examined is shown in Figure 3-5B, where the origin is the back bottom left corner of the cube. From this figure, we can see that a sizable portion of the solution space can be discarded. In regions where there are both viable and redundant solutions (for example, quadrants 2 and 3 in Figure 3-5A), the addition of constraints requiring that θ1 ≤ . . . ≤ θk in the problem of maximizing the expected improvement ensure that only the unique portion of the region is considered. If more than one non-coplanar orientation is optimized, a similar symmetry to the multiple coplanar orientation symmetry exists. Consider an implementation where two non-coplanar beam orientations are optimized, and these orientations are obtained from rotating both the gantry and the couch. Each beam is represented by two variables in the solution vector: one variable indicating the degree of gantry rotation, and one variable indicating the degree of couch rotation. Let θ1 and θ2 be the gantry rotation and couch rotation of the first beam, respectively, and θ3 and θ4 be the gantry rotation and couch rotation of the second beam, respectively. Then, the solution vector {θ1 , θ2 , θ3 , θ4 }

68

is identical to the solution vector {θ3 , θ4 , θ1 , θ2 }. Because the couch angle selected is dependent on the gantry angle (and vice versa), this symmetry can be exploited by only removing redundant solutions from one of the beam variables, that is, by requiring that θ1 ≤ θ3 (removing redundancy from the gantry angles) or θ2 ≤ θ4 (removing redundancy from the couch angles). In general, if d degrees of motion are used to obtain m beam orientations, and the linear accelerator motion variables are in the same order for each beam, then θk ≤ θk+d ≤ θk+2d ≤ . . . ≤ θk+(m−1)d for some k ∈ {1, . . . , d}. 3.6.3

Method of Obtaining the Next Observation

The RS algorithm allows for two methods of selecting the next point to observe: by maximizing the expected improvement or by maximizing the uncertainty. In these tests, the point to observe is obtained by first selecting the point that maximizes the expected improvement until the maximum expected improvement falls below a certain threshold, and then switching to the point that maximizes the uncertainty. Once the maximum uncertainty also falls below a certain threshold, the algorithm terminates. By first selecting according to the expected improvement, the method quickly obtains a good solution. By then selecting according to uncertainty, theoretical convergence to the global minimum is ensured. 3.7 3.7.1

Neighborhood Search

Introduction

From Aleman et al. [79], we test the simulated annealing algorithm on the BOO problem, as well as existing and new variants of a greedy neighborhood search heuristic called the Add/Drop algorithm (see Kumar [80]) to obtain a good solution to the BOO problem. In each step of the Add/Drop algorithm, a beam in the current beam set is replaced by a neighboring beam that yields an improving solution. As with the simulated annealing implementation, we also apply our new neighborhood to the Add/Drop algorithm and compare its performance to a commonly used neighborhood structure.

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3.7.2

Neighborhood Search Approaches

Neighborhood search approaches are common methods of obtaining solutions to global optimization problems. For a vector of decision variables, a neighbor is obtained by perturbing one or more of the decision variables. A neighborhood for a particular vector of decision variables is the set of all its neighbors for a given method of perturbating the decision variable vector. A solution is considered to be locally optimal if there are no improving solutions in its neighborhood. Both deterministic and stochastic neighborhood search algorithms have been applied to a wide variety of optimization problems. A deterministic neighborhood search algorithm is one in which the entire neighborhood, or a pre-defined subset of the neighborhood, is enumerated in each iteration to find an improving solution. Stochastic versions of neighborhood search approaches, for example, simulated annealing, randomly select neighboring solutions in an attempt to find an improving solution in each iteration. For the BOO problem, we consider two neighborhood search methods. The first is a deterministic neighborhood search algorithm that finds a locally optimal solution, and the second is the simulated annealing algorithm, which, although based on neighborhood searches, provably converges to the globally optimal solution for certain neighborhood structures. 3.7.3

A Deterministic Neighborhood Search Method for BOO

Deterministic neighborhood search methods are optimization algorithms that start from a given solution and then iteratively select the best point in the current neighborhood as the next iterate. The best point in the neighborhood can be found by complete enumeration if the neighborhood is small, or by optimization is the neighborhood is large or if objective function evaluations are expensive. Due to the complexity of the BOO problem, even when only a subset of available orientations is considered, we will focus on smaller neighborhoods and use enumeration. The neighborhood could alternatively be searched heuristically, for example by searching the neighborhood until

70

the first improving solution is found, rather than the best improving solution. If no improved solution can be found the current solution is a local optimum. In our implementation of the Add/Drop algorithm, a small neighborhood is desired for enumeration purposes. In each iteration, a neighborhood for just a single beam is considered. Say a beam set consisting of k beams is desired. Letting the neighborhood of a single beam θh in θ be denoted as Nh (θ), the Add/Drop algorithm is as follows: • Initialization: 1. Choose an initial starting solution θ (0) . 2. Set θ ∗ = θ (0) and i = 0. • Iteration: ¯ ∈ Nh (θ (i) ). 1. Select h ∈ {1, . . . , k}, then generate θ ¯ < F (θ ∗ ), set θ ∗ = θ (i+1) = θ ¯ and set i ← i + 1. 2. If F (θ) 3. If all points in ∪kh=1 Nh (θ (i) ) have been sampled without improvement, stop with θ ∗ as a local minimum. Otherwise, repeat Step 1. 3.7.3.1

Neighborhood Definition

In each step of the Add/Drop algorithm, a beam in the current solution is replaced with an improving beam in its neighborhood. Rather than define a neighbor as related to an entire beam vector, the neighborhoods of individual beams are considered. The neighborhood of a single beam θh in θ is defined as Nh (θ) =

n

(θ1 , . . . , θh−1 , θ mod 360, θh+1 , . . . , θk ) o ∈ B k : θh − δ ≤ θ ≤ θh + δ .

In other words, the neighborhood of a beam is all beams within ± δ degrees taking into account the cyclic nature of the angles. The cyclicality of the angles refers to the fact that all angles can be represented by degrees in [0,360]. For example, 400◦ = 40◦ and −100◦ = 260◦ . The expression θ mod 360 captures this cyclicality.

71

3.7.3.2

Neighbor Selection

The process of selecting a neighboring point in each iteration consists of two steps: selecting the index h to change and then selecting an improving angle in Nh (θ) to replace θh . If h is selected as i mod k + 1, the algorithm will cycle through each index sequentially, similar to a Gibbs Sampler (see, for example, Geman and Geman [81] and Gelfand and Smith [82]). The Gibbs Sampler also uses a similar two-step approach to generating a new point by sequentially generating a new value for each variable in turn. If h is selected randomly in each iteration, the resulting algorithm is similar to a Hit-and-Run method (see, for example, Smith [83] and Bélisle [84]), in which a variable to be changed is selected randomly, and then a new value for that variable is also selected randomly within a neighborhood. Once h is selected, the new value for θh can be generated by enumeration or by a heuristic method. The Add/Drop algorithm compares the quality of the new solution to the current solution, and then only accepts improving solutions. This greedy approach results in a locally optimal solution. 3.7.3.3

Implementation

The index of beam angle to be changed in each iteration, h in Step 1 of the algorithm in Section 3.7.3, is chosen as h = i mod k + 1 to cycle through each index in a sequential ¯ in iteration i is manner. In the Add/Drop implementation, once h is determined, θ ¯ = arg min chosen as θ θ∈Nh (θ (i) ) {F (θ)}. By replacing each beam by the most improving neighbor, the Add/Drop algorithm is a greedy heuristic which terminates when there is no improving neighbor for any beam. A multi-start aspect is added by repeating the algorithm with multiple initial starting points. For example, one strategy to select starting points would be to select a random starting point according to a particular distribution. Another strategy would be to select an equi-spaced solution and rotate it a fixed number of times to obtain new starting points until the initial equi-spaced solution is repeated. Equi-spaced beam solutions are common

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in clinical practice for an odd number of beams. The reason that such a method is not generally used in practice for even-numbered beams is that the resulting beam set would contain parallel-opposed beams (beams that lie on the same line), which are not used by convention as it is believed that the effect of a parallel-opposed beam is very similar to simply doubling the radiation delivered from a beam. If an equi-spaced solution is not possible given a beam set of k beams and the discretization level of the candidate beam (0)

set B, then the solution can be rounded so that θ h ∈ B, h = 1, . . . , k. 3.7.4

Simulated Annealing

The simulated annealing algorithm used is similar to the classical simulated annealing approach proposed in Kirkpatrick et al. [85]. The simulated annealing algorithm is based on the Metropolis algorithm, wherein a neighboring solution to the current iterate is generated, and if it is an improving point, it becomes the current iterate. Otherwise, it becomes the current iterate with probability exp{∆F/T }, where ∆F is the difference in FMO value between the current iterate and the newly generated point and T is the temperature, a measure of the randomness of the algorithm. If T = 0, then only improving points are selected. If T is very large, then any move is accepted, which is essentially a random search. The simulated annealing algorithm starts with an initial temperature T0 and performs a number of iterations of the Metropolis algorithm using T = T0 . Then, the temperature is decreased according to some cooling schedule such that {Ti } → 0. Obvious parallels can be drawn between the simulated annealing algorithm and the Add/Drop neighborhood search method described in Section 3.7.3. While the Add/Drop algorithm deterministically searches the neighborhood for improving solutions, the simulated annealing algorithm randomly selects neighboring solutions. Rather than being limited by the ability to only move to improving solutions, the simulated annealing algorithm may still move to a non-improving solution with a certain probability, thus

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allowing for the escape from local minima. The Add/Drop algorithm, on the other hand, is a greedy algorithm that is specifically designed to find local minima. The simulated annealing algorithm is essentially a randomization of the Add/Drop algorithm. In addition to the added randomness, the possibility of changing more than one beam in each iteration is allowed by selecting a set of indices H ⊆ {1, . . . , k} to change, rather than just selecting a single index h. The simulated annealing algorithm is as follows: • Initialization: 1. Choose an initial beam set θ (0) and calculate its FMO objective function value F0 . ˆ = θ (0) , Fˆ = F0 , i = 0. 2. Set θ • Iteration: 1. Select H ⊆ {1, . . . , k}, generate θ ∈ ∪h∈H Nh (θ (i) ), and calculate its FMO objective function value F . ˆ = θ. Otherwise, set Fi+1 = F 2. If F < Fˆ , set Fˆ = F , Fi+1 = F , θ (i+1) = θ and θ (i+1) and θ = θ with probability exp{(Fi − F )/Ti }. 3. Set i ← i + 1 and repeat Step 1. The simulated annealing algorithm has been previously applied to the BOO problem. Bortfeld and Schlegel [35] use the “fast” simulated annealing algorithm described by Szu and Hartley [86] which employs a Cauchy distribution in generating neighboring points. Stein et al. [40], Rowbottom [39] and Djajaputra et al. [36] also use a Cauchy distribution in generating neighoring solutions. Lu et al. [37] randomly select new points satisfying BEV and conventional wisdom criteria and Pugachev and Xing [38] randomly generate new points and then vary them according to an exponential distribution. All accept improving solutions, and with the exception of Rowbottom et al. [39] who only accept improving solutions (essentially Ti = 0 for all i), all accept non-improving solutions with a

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Boltzmann probability. None of the previous BOO studies employing simulated annealing use the exact FMO as a measure of the quality of a beam set. 3.7.4.1

Neighborhood Definition

Two neighborhood structures are explored. The first neighborhood is similar to that described in Section 3.7.3.1 in that a neighborhood Nh (θ) is considered for only a single beam index h ∈ {1, . . . , k}, just as in the Add/Drop method. As an extension to changining a single angle in each iteration, we also consider a neighborhood that involves changing all beams in each iteration, corresponding to H = {1, . . . , k} in Step 1 of the simulated annealing algorithm in Section 3.7.4. This neighborhood is defined as N (θ) = ∪kh=1 Nh (θ). Again, the neighborhoods for the individual beams are defined as in the first method, with bounds of ± δ degrees. 3.7.4.2

Neighbor Selection

The method of selecting a neighbor depends on the neighborhood structure as described in Section 3.7.4.1. In the first method where only one beam is changed at a time, a neighbor is selected using the randomized approach described in Section 3.7.3.2. Once h is selected, the probability of selecting a particular solution in Nh (θ) where the new θ is d degrees from θh is P {D = d}, where D is the realization of a random variable of some probability distribution defined on the interval [−δ, −δ + 1, . . . , δ]. For the neighborhood N (θ) where all beams are changed in an iteration, the new value for each beam h ∈ {1, . . . , k} is generated from Nh (θ) in the same manner described above. 3.7.4.3

Implementation

In addition to basing our algorithms on the exact FMO solution rather than on heuristics or scoring measures, our simulated annealing approach differs from the previous studies in the distribution used to generate neighbors, the definition of the neighborhood, the cooling schedule and the number of iterations/restarts used. Not only do we use a new neighborhood structure, but also a geometric probability distribution rather than a

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uniform or Cauchy distribution on the neighborhood. The geometric distribution is similar in shape to the Cauchy distribution in that they both can have fat tails depending on the choice of probability parameters. The fat tails of these distributions allow for points far away from the current solution to be selected as successive iterates, which potentially increases the likelihood of finding a globally optimal solution. The geometric distribution has the added attractiveness of producing discrete solutions, which is desirable for the BOO problem in which discrete solutions are preferred. By using the cooling schedule Ti+1 = αTi with α < 1, the sequence of temperatures {Ti } converges to zero as the number of iterations increases. In our approach, the neighborhood of a beam for both the Nh (θ) and N (θ) neighborhoods is defined using δ = 180, that is, Nh (θ) = B. By defining the neighborhood of each beam to be the entire single-beam solution space, the simulated annealing algorithm converges to the global optimum when using the neighborhood N (θ) defined in Section 3.7.4.1. Though Nh (θ) is large, each beam in Nh (θ) is assigned a probability so that only the beams closest to θh have a significant probability of being selected. Figure 3-7A shows the probability of replacing θh with beams at varying distances using probability p = 0.25 for the geometric distribution. Note that the current beam cannot be selected as a replacement. As with the Add/Drop method, a multi-start aspect is added to the simulated annealing algorithm by repeating the algorithm using several different starting points. 3.7.4.4

Convergence

Unlike many previously proposed simulated annealing algorithms, our algorithm converges to the globally optimal solution to the BOO problem under mild conditions. The following theorem summarizes these conditions. Theorem 3.7.1. Suppose that • H = {1, . . . , k} • limi→∞ Ti = 0 • δ = 180

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• There is a positive probability of generating any solution in the neighborhood. Then our simulated annealing algorithm converges to the global optimum solution in the sense that lim Fi = F ∗

in probability

i→∞

where F ∗ is the global optimum value of the BOO problem. Proof. This follows from Theorem 1 in Bélisle et al. [87]. 3.7.5

A New Neighborhood Structure

For the BOO problem, the neighborhood structure that is typically used for a vector of beam orientations is simply the collection of beam vectors obtained from changing one or more of the beams to a neighboring beam, where each beam has its own neighborhood Nh (θ). In addition to Nh (θ), we consider a new neighborhood which we call a “flip” neighborhood. The flip neighborhood of a beam index h consists of Nh (θ) plus a neighborhood around the parallel opposed beam of h. The parallel opposed beam is the beam 180◦ away, that is, h0 = (θh + 180) mod 360 The flip neighborhood can be defined as NhF (θ) =

n

(θ1 , . . . , θh−1 , θ mod 360, θh+1 , . . . , θk ) ∈ B k o : θ ∈ [θh − δ, θh + δ] ∪ θh + 180 − δ F , θ + 180 + δ F

Note that the values δ and δ F may be different. Figure 3-6 depicts a flip neighborhood for a beam located at 0◦ degrees, the center of the top shaded wedge representing Nh (θ), where θh = 0. The motivation for the flip neighborhoods arises from the observation that many of the 3-beam simulated annealing plans generated using the regular neighborhood contained two beams very close to two beams in the optimal solution (obtained by explicit

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enumeration), while the third beam was very close to the parallel opposed beam of the third beam in the optimal solution. Given this observation, it is intuitive that the inclusion of the neighborhood around the parallel beam should provide improved solutions. The neighborhoods Nh (θ) and NhF (θ) with varying δ F values are applied to both the Add/Drop and the simulated annealing frameworks. For the geometric probability distribution used in the simulated annealing method, Figure 3-7B shows the probability of selecting beams at different distances using a flip neighborhood with probability p = 0.25. Note that the current beam cannot be selected as its own neighbor.

Figure 3-6. Nh (θ) (top shaded area) and NhF (θ) (top and bottom shaded areas) for θh =0.

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3.8

Results

In addition to judging the BOO algorithms according to their computational time, the plans must also be evaluated for clinical viability. All criteria used are those employed at the Davis Cancer Center at Shands Hospital at the University of Florida. 3.8.1

Evaluating Plan Quality

In order to formulate an optimization problem, a quantitative measure of the treatment plan quality is needed. This measure, the FMO function value, needs to appropriately make the trade-off between the contradictory goals of covering targets and sparing critical structures. Typically, a good plan ensures that at least a certain percent of each target receives the prescription dose. A coldspot occurs where less than a certain percent of the target receives the prescription dose. Similarly, a hotspot occurs if a significant percentage of the target receives more than the prescription dose. 3.8.1.1

Target coverage

Each of the plans contains two target structures, or planning tumor volumes (PTV): one is the tumor mass observed from imaging scans, which we will call PTV2, and the other is the PTV2 plus some margin specified by the physician, which we will call PTV1. The PTV1 structure is used by physicians in case there are elements of the tumor mass that cannot be seen from the imaging scans. The dose prescribed for PTV1 is less than the dose prescribed for PTV2. For target structures, we require that at least 95% of the target receives the full prescription dose, so the dose that is received by at least 95% of each of the targets is measured. We want to restrict the amount of the target that receives more than the prescription dose. Because PTV2 is contained inside PTV1, PTV2 will necessarily have a sizable, but less important, area receiving an overdose. Thus, we are only concerned with PTV2 overdose. To evaluate the size of the hotspot, we check the percent volume of PTV2 that receives more than 110% of the prescription dose. To evaluate the coldspots, we check

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Table 3-1. Sparing criteria varies for each critical structure Structure Percent (%) ≤ Dose (Gy) brain stem 100 55 eyes 50 30 mandible 100 70 optic chiasm 100 55 optic nerves 100 50 parotid glands 50 30 skin 100 60 spinal cord 100 45 submandibular glands 50 30 the percent volume of both PTV1 and PTV2 that receives at least 93% of the prescription dose. The prescription doses are set to 54 Gy for PTV1 and 73.8 Gy for PTV2, which are the dose values used at Shands Hospital at the University of Florida. 3.8.1.2

Critical structure sparing

The critical structures involved in each case vary, depending on their proximity to the tumor. The critical structures can be classified into two general groups according to their ability to survive radiation dose. Parallel structures, e.g., saliva glands, will continue to function as long a certain percentage of the organ receives less than a certain amount of dose. Serial structures, on the other hand, will cease to function if any of the organ receives over a certain amount of dose. The spinal cord is one example of a serial structure—if it receives too much dose, the effect is equivalent to cutting it in half, leaving the patient paralyzed. The sparing criteria for each of the common critical structures in head-and-neck cases are listed in Table 4-2. The critical structures involved in each case vary, depending on their proximity to the tumor. There are four saliva glands: one submandibular and one parotid gland on each of the right and left sides. The saliva glands are of particular importance because their loss can greatly decrease the patient’s quality of life, but because of their location relative to the usual tumor positions, they can be difficult to spare. Studies show that a patient can lead a relatively normal life with three of the four glands spared. The loss of other

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organs, especially the spinal cord, will also greatly affect the patient’s quality of life, but head-and-neck tumors are usually situated in such a way that other organs can be easily spared in the FMO optimzation. Thus, the results presented place particular emphasis on the sparing of saliva glands. Rather than relying strictly on FMO value, a tool commonly used by physicians to judge the quality of a treatment plan is the dose-volume histogram (DVH). This histogram is a measure of the cumulative dose received by a given structure. It specifies the fraction of each structure’s volume that receives at least a certain amount of dose. Although there are several critical structures to be considered in head-and-neck cases, the saliva glands are notoriously the most difficult to spare due to their proximity to common tumor locations. Thus, for clarity, the DVH results provided include only target structures and saliva glands. Each of the treatment plans spares all organs not shown in the DVHs. In the DVH results provided, vertical lines indicate target prescription doses, and asterisks mark the sparing criteria for the saliva glands. 3.8.2

Response Surface Method Results

The response surface method was tested on six head-and-neck cases using a Windows XP computer with a 3.2 GHz Pentium IV processor and 2 GB of RAM. The sizes of the test cases for plans with three beams are shown in Table 3-2. Each algorithm was allowed to run for 12 hours, which is not an unreasonable run length because BOO will not be performed on a day-to-day basis. It is anticipated that BOO will be performed once overnight between the time the patient is imaged and the time the patient begins radiation therapy. A good beam vector chosen before treatment begins should continue to provide quality treatment plans throughout the patient’s treatment, which is typically 35 days. The beam orientations from which linear accelerators are capable of delivering radiation are not restricted to integer value degrees. In this study, integral beam orientations are desired to account for setup tolerances. For the same reasons, beam orientations are considered on a 10◦ grid. To obtain integral solutions, in the subproblem

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Table 3-2. Sizes of test cases. Case 1 2 3 4 5 6 Avg. Low High

# bixels # voxels 514 345,629 546 352,284 613 347,233 549 268,823 423 271,156 585 389,565 538 329,115 423 268,823 613 389,565

of maximizing I(θ), the integer constraint is relaxed in the problem of determining an upper bound on s2 (θ), and the resulting solution is rounded to integer values. The branching scheme used treats the rounded solution as integral and branches so as to avoid overlapping subregions as described in Section 3.6.2.3. Results are provided for each possible initial region scheme. The point at which branching is performed in each region, θ b in Section 3.6.2.3, is chosen as the midpoint of the region. Also, r¯i and θ¯h in the underestimating terms in Problem s2 -UB in Section 3.6.2.2 are taken to be the midpoints of their respective intervals. It is anticipated that the weighted distance measure in equation 3–3 will have an significant impact on the algorithm’s performance. Intuitively, a small weighted distance corresponds to a small correlation between points, which will cause the algorithm to behave locally. In order to induce the algorithm to behave globally, the algorithm must assume less correlation between two points. If the points are less correlated, the algorithm will be less likely to stay in the neighborhood of previously sampled points. The correlation between two points can be decreased by increasing the weighted distance between the points, which can be done by increasing c or p. If c becomes sufficiently large, the correlation between points will be effectively zero, thus yielding an effectively random search algorithm. To test these expectations, c was tested with values of 10.0, 100.0 and

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500.0. In each test, five randomly selected starting points were used to initialize the RS algorithm. To evaluate the algorithm’s performance across all of the tested cases, the relative improvements in FMO value over a 5-beam equispaced plan (denoted 5 equi ), a 7-beam equispaced plan (denoted 7 equi ) and a locally optimal 3-beam coplanar plan obtained using a local search algorithm called the Add/Drop local search heuristic introduced by Kumar [80] and denoted 3 A/D are compared. 3.8.2.1

Proof of concept

To test the accuracy of the RS method, a single case was tested wherein the problem of adding a single coplanar beam to an equi-spaced, coplanar 3-beam solution over a 1◦ grid was considered. The algorithm was initialized with two randomly selected starting points. By considering such a small scale problem, the solution space in each iteration can be explicitly enumerated in order to exactly obtain the next best point to sample. The ability to enumerate the solution space will also allow us to determine how accurately the RS method models the FMO objective function. At each point that has been sampled, both the uncertainty and the expected improvement will be zero. This result is not only theoretically true, but also intuitive because once the FMO value at a certain point is known, there will be no improvement over the current best FMO value by sampling that point again. It is also expected that as the algorithm progresses, the approximation of the FMO function will become increasingly accurate, with the approximation obtaining the exact FMO values at sampled points. Figures 3-8A-3-8D demonstrate how the RS method behaved as predicted at different points in the RS algorithm. The expected value is zero at sampled points and the approximation of the FMO function almost perfectly fits the true FMO function by the time the algorithm terminates. The importance of the starting points, the points sampled before the algorithm begins to give the method some baseline information about the FMO function, was also tested.

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Figure 3-8. Proof of concept results at various stages of the RS algorithm. A) After two points. B) After 20 points. C) After 80 points. D) After 148 points, when the algorithm terminates. The RS method was run with 100 randomly generated sets of starting points, and the RS method obtained the global optimum in 90.6% of trials, indicating that the performance of the algorithm is not significantly dependent on the starting points. 3.8.2.2

Adding a non-coplanar beam to a coplanar solution

Next, the problem of adding a non-coplanar beam to a 3-beam locally optimal coplanar solution was considered. The locally optimal solution is obtained using the Add/Drop algorithm. The beam data for the non-coplanar beam being optimized is generated on-the-fly, and consists of gantry and couch rotations, where the both gantry and couch are allowed to rotate a full 360◦ on a 10◦ grid. As the final solution of the non-coplanar RS plan will be a 4-beam plan, the results from the response surface solution

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are compared to the locally optimal coplanar 4-beam Add/Drop plan, denoted 4 A/D. The plans will also be compared to an equi-spaced, coplanar 7-beam plan, denoted 7 equi, which is commonly used in practice to treat head-and-neck cancers. There is relatively little deviation in the final solutions between the different parameter choices and initial regions schemes, as shown by Table 3-3. The results also indicate that the starting points chosen do not significantly affect the outcome of the algorithm. This implies that the response surface algorithm is robust with respect to varying implementations. Although the 4 RS solutions obtained an average of 5.44% decrease in FMO value from the 7 equi plans, the 4 RS solutions did in fact obtain an average of 16.12% improvement in FMO value over the 4 AD solutions. Despite the differences in FMO value, all treatment plans examined were similar in clinical quality, as discussed in Section 3.8.2.3. Although the algorithm was allowed to run for 12 hours in each scenario, the minimum FMO value obtained by the RS method was found early on. On average, the best FMO value found was obtained in 6.15 hours after sampling 27-40 points. For each of the RS method variations tested, both the number of points sampled and the relative improvements in FMO value are nearly identical. This indicates that the algorithm is robust with respect to parameter and implementation changes. The time spent generating beam data comprises approximately 84% of the algorithm’s run time, while the response surface portion on average accounts for only 13%. Thus, it is expected that changes to the RS method, including improvements to the branch-and-bound routine, will not have a very strong impact on the number of points the algorithm will sample in its allotted run time. 3.8.2.3

Clinical results

The target coverage achieved by the different treatment plans are displayed in Table 3-4. On average, the 7 equi plan was able to deliver the most amount of dose to PTV2,

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Table 3-3. Minimum FMO value obtained and time required to obtain it. Min. FMO value Time (hrs) Case Avg. St. Dev. Avg. St. Dev. 1 565.24 8.82 5.35 5.07 2 570.51 12.83 7.49 3.78 3 927.34 20.60 7.05 2.21 4 710.92 7.72 6.54 3.39 5 512.22 20.04 6.96 3.33 6 799.95 34.07 3.48 3.40 Table 3-4. Target coverage achieved by the treatment plans 4 RS 4 A/D PTV2 dose at 95% volume 73.16 Gy 72.56 Gy PTV2 % receiving > 110% of Rx 23.18 % 15.07 % PTV2 % receiving > 93% of Rx 98.87 % 98.67 % PTV1 dose at 95% volume 54.71 Gy 54.41 Gy PTV1 % receiving > 93% of Rx 97.95 % 98.01 %

7 equi 73.81 Gy 31.63 % 99.57 % 55.09 Gy 97.46 %

but the 4 RS plan is very close. Both of the 4-beam plans obtain smaller hotspots and better PTV1 target coverage than the 7 equi plan. The 4 A/D plan on average underdoses PTV2, which could lead to recurrence of the cancer. This underdosage could also account for the smaller hotspot in the 4 A/D plans. Figures 3-10 and ?? illustrate two representative cases where the 4 RS, 4 A/D and 7 equi plans each have clinically acceptable target coverage. The vertical line at 73.8 Gy indicates the prescription dose for PTV2. The ability of each of the treatment plans to spare the organs in the cases tested is shown in Table 3-5. Surprisingly, both the 4 RS and the 4 A/D plan are equivalent to or outperform the 7 equi plan in terms of organ sparing. In the 4-beam plans, the left submandibular gland is spared in 83% of the treatment plans developed, whereas it is only spared in 67% of the 7 equi plans. One case illustrating equivalent organ sparing is shown in Figure 3-10, and one case demonstrating improved organ sparing over the 7 equi plan is shown in Figure ??. Just as PTV2 underdosage in the 4 A/D plans likely contributed to the smaller hotspots, it is possible that the improved organ sparing in the 4 A/D plans is also a result of the underdosage.

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Table 3-5. Percentage of plans in which an Structure brain stem mandible left optic nerve right optic nerve left eye right eye optic chiasm left parotid gland right parotid gland left SMB gland right SMB gland spinal cord skin

organ is 4 RS 100% 100% 100% 100% 100% 100% 100% 100% 67% 83% 50% 100% 100%

spared. 4 A/D 7 equi 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 67% 67% 83% 67% 50% 50% 100% 100% 100% 100%

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Figure 3-9. 7-beam equi-spaced (dotted), 4-beam Add/Drop (dashed) and 4-beam RS non-coplanar (solid) target and select saliva gland DVHs. A) Target coverage is nearly identical. B) The tumor surrounds the right submandibular gland, so the FMO solver recognizes that it cannot be spared and allows it to receive as much dose as necessary to ensure good target coverage in all plans. All other saliva glands are spared in all plans.

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Figure 3-10. 7-beam equi-spaced (dotted), 4-beam Add/Drop (dashed) and 4-beam RS non-coplanar (solid) target and select saliva gland DVHs. A) Target coverage is nearly identical. B) The left submandibular gland is spared by the two 4-beam plans, but not by the 7-beam plan. All other saliva glands are spared in all plans. 3.8.3

Neighborhood Search Method Results

The simulated annealing method was tested on six head-and-neck cases using a Windows XP computer with a 2.13 GHz Pentium M processor and 2 GB of RAM. On average, ≈ 340 FMOs were calculated in 30-minute run time allowed for the simulated annealing and Add/Drop algorithms. Beams were selected on a 5-degree grid, yielding 72 candidate coplanar beams. The simulated annealing and Add/Drop algorithms were used to obtain 4-beam coplanar plans using regular and flip neighborhoods. In order to compare the quality of the treatment across different plans, the plans are compared in terms of the percentage improvements of each plan’s FMO value improvement over the FMO value of the locally optimal 3-beam plan obtained from the Add/Drop local search heuristic described by Kumar [80]. The Add/Drop plans are denoted 3 A/D and 4 A/D for the 3- and 4-beam plans, respectively. The 4-beam plans generated by the simulated annealing and Add/Drop algorithms are compared to the typical 5- and 7-beam equi-spaced plans,

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denoted 5 equi and 7 equi, respectively. The simulated annealing plans are denoted by the implementation numbers, which refer to the parameters used, given in Table 3-6. Figure 3-11 demonstrates the improved convergence times possible using the flip neighborhood. 3.8.3.1

Add/Drop algorithm results

The Add/Drop algorithm was allowed to run for 30 minutes to generate a 4-beam plan. The Nh (θ) neighborhood with δ = 20 and the NhF (θ) with δ F = 0 and δ F = 20 neighborhoods are tested for the Add/Drop algorithm. The value δ = 20 is chosen to approximate the neighborhood size that is expected from the simulated annealing implementation using a large flip neighborhood, where δ F = 180. More details on the simulated annealing implementations are provided in Section 3.8.3.2. Using Nh (θ), the 4-beam Add/Drop solution is nearly identical identical to the 7-beam equi-spaced plan, while the flip neighborhoods allow the Add/Drop algorithm to find 4-beam solutions that exceed the quality of the 7-beam plans. Figure 3-12 demonstrates the quality of the solutions, while Figure 3-11(a) illustrates that the flip neighborhoods provide faster FMO convergence than that of Nh (θ). 3.8.3.2

Simulated Annealing results

Several parameter sets were tested for the simulated annealing algorithm. For simplicity, each of the parameter sets and methods of generating a neighboring solution are numbered according to Table 3-6. Each implementation contains a total of 500 iterations, i.e., 500 sampled points, thus yielding a fair comparison between the parameters. To ensure clinical practicality, the algorithm was allowed to run for a maximum of 30 minutes or 500 iterations, whichever came first. For the cooling schedule, we update the temperature according to an exponential cooling schedule, Ti+1 = αTi , where α < 1. Due to the random nature of the algorithm, the algorithm is restarted five times, each time with a different initial starting point. The first initial starting point is an equi-spaced solution, and each subsequent starting point is

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600 500

550 0

5

10

15 20 time (minutes)

25

30

450 0

A

5

10

15 20 time (minutes)

25

B

Figure 3-11. Comparison of FMO convergence. A) Add/Drop. B) Simulated annealing. the previous initial solution rotated by d degrees, where candidate angles are considered on a d-degree grid, that is, every dth angle is considered. The number of simulated annealing and Metropolis iterations are chosen such that the total number of iterations is 500. The initial temperature values tested are T0 = 0 and T0 = 75. T0 = 0 results in the acceptance of only improving solutions, while the initial temperature value 75 was selected as the value that would approximately yield a 50 percent probability of selecting a non-improving solution for the initial iterations of the algorithm. For both the Nh (θ) and NhF (θ) neighborhoods, δ = δ F = 180 is used so that the entire solution space is considered as a neighborhood. As shown in Figure 3-7A, the probability of selecting a beam 20◦ away using the Nh (θ) neighborhood with geometric distribution with p = 0.25 is only 0.39% on a 5◦ grid. We consider this sufficiently small to not consider neighborhoods larger than δ = 20 for Nh (θ) and δ F = 20 for NhF (θ) in the Add/Drop algorithm. Just as in the Add/Drop implementation, the neighborhood NhF (θ) with δ F = 0 is also considered.

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30

Table 3-6. Definitions of implementations. Number n 1 100 2 10 3 100 4 10 5 100 6 10 7 100 8 10 9 100 10 10 11 100 12 10 13 100 14 10 15 100 16 10

m 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10

N 1 1 1 1 1 1 1 1 all all all all all all all all

α 0.9 0.9 0.99 0.99 0.9 0.9 0.99 0.99 0.9 0.9 0.99 0.99 0.9 0.9 0.99 0.99

T0 0 0 0 0 75 75 75 75 . 0 0 0 0 75 75 75 75

Figure 3-11(b) shows that the flip neighborhoods converge in FMO value significantly faster than does the Nh (θ) neighborhood, while Figure 3-13 shows that the flip neighborhoods provide comparable solution quality to both the non-flip simulated annealing and 7-beam equi-spaced solutions. 3.8.3.3

Clinical results

Because there is no fundamental way of quantifying a treatment plan, a tool commonly used by physicians to judge the quality of a treatment plan is the dose-volume histogram (DVH). A DVH is a graphical measure of the cumulative dose received by a given structure. It specifies the percentage of each structure’s volume that receives at least a certain amount of dose, thus providing an intuitive means of assessing the quality of a treatment plan. The plans tested plans each contain two target structures. The gross tumor volume (GTV) is the tumor mass observed from imaging scans. The clinical tumor volume (CTV) is the GTV plus some margin specified by the physician. The CTV is used by physicians

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in case there are elements of the tumor mass that cannot be seen from the imaging scans, and the dose prescribed for the CTV is less than the dose prescribed by the GTV. DVHs for a representative case comparing the 7-beam equi-spaced plan with the simulated annealing plans obtained using a regular neighborhood and flip neighborhoods with δ F = 0 and δ F = 180 are shown in Figure 3-13. Comparison of the 7-beam equi-spaced plan and the Add/Drop plans using a regular neighborhood and flip neighborhoods with δ F = 0 and δ F = 20 are shown in Figure 3-12. The sparing criteria used for the saliva glands, no more than 50% of the gland receiving 30 Gy, is marked by the star in Figures 3-13 and 3-12. The prescription dose for the GTV is 73.8 Gy, which is marked by the vertical line in Figures 3-13 and 3-12. As previously stated, for target structures, we require that at least 95% of the target receives the full prescription dose. Figure 3-13 reveals that the 7-beam equi-spaced plan actually overdoses the target and has a larger hotspot than the 4-beam simulated annealing plans. The 7-beam equi-spaced plan only spares three of the four saliva glands, whereas the 4-beam simulated annealing plans spare three or more saliva glands. The simulated annealing plans obtained using the flip neighborhoods spare all four saliva glands, while the plan obtained how the Nh (θ) neighborhood only spares three saliva glands, indicating that the flip neighborhoods do in fact find superior solutions in terms of clinical quality. Figure 3-12 shows that the 4-beam Add/Drop plans obtain nearly identical solutions when compared to the 7-beam equi-spaced DVHs. The flip neighborhoods perform clinically comparably to the regular neighborhood plans, and all of the Add/Drop plans are comparable to the 7-beam equi-spaced plan in terms of saliva gland sparing and target coverage. 3.9 3.9.1

Conclusions and Future Directions

Response Surface Conclusions

We have shown that for head-and-neck cases, quality plans with fewer beams than a standard treatment plan can be obtained if BOO is applied. The response surface

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Add/Drop: Target DVHs

Add/Drop: Saliva gland DVHs

100

7−beam equi−spaced flip, δF=20

80

flip, δ =0 Regular neighborhood


100

flip, δF=0 Regular neighborhood

60 40

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F

20 0 0

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Figure 3-12. Comparison of Add/Drop and 7-beam equi-spaced plans. A) The Add/Drop plans achieve nearly identical target coverage when compared to the 7-beam equi-spaced plan. B) The saliva gland sparing in the Add/Drop plans and the 7-beam equi-spaced plan is clinically equivalent.

Simulated Annealing: Target DVHs


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Figure 3-13. Comparison of Add/Drop and 7-beam equi-spaced plans. A) Unlike the 7-beam equi-spaced plan, the 4-beam simulated annealing plans do not overdose the target. B) The simulated annealing plans are also capable of sparing more saliva glands than the 7-beam equi-spaced plan.

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algorithm operates in a clinically reasonable time frame, and is generally successful in selecting non-coplanar beam orientations to improve the FMO value over that of locally optimal coplanar solutions. The FMO value of the 4-beam response surface plans was also only slightly larger than that of the 7-beam equi-spaced coplanar treatment plans, indicating comparable treatment plans despite the decrease in the number of beams used. In terms of clinical results, the most significant benefit of the non-coplanar solutions over the locally optimal coplanar solutions was the ability to deliver a higher amount of dose to the target structures. Both the non-coplanar and locally optimal coplanar solutions were able to obtain treatment plans with organ sparing that is comparable to or improved upon the 7-beam equi-spaced coplanar treatment plans. While the inclusion of non-coplanar orientations in BOO is useful in terms of FMO value and target coverage, the resulting improvements in the treatment plan may not always be clinically significant. With better parameter tuning or neighborhood structure, it is possible that the Add/Drop algorithm can obtain coplanar treatment plans with more desirable target coverage, thus making the response surface plans and the Add/Drop plans clinically equivalent. This suggests that the inclusion of non-coplanar beam orientations does not significantly improve the quality of a treatment plan. Although most BOO research is restricted to coplanar orientations, there has not yet been a study assessing the solution quality of coplanar versus non-coplanar solutions. With this study as evidence, both researchers and practioners now have a basis for restricting the solution space to the smaller, more tractable set of coplanar beams for head-and-neck beam optimization. The patient cases in this work were all head-and-neck cases. Different tumor sites, e.g., breast, lung and prostate, could also benefit from BOO, and perhaps may experience greater improvements in treatment plan quality. In future work, these sites will be tested to assess the general clinical usefulness of non-coplanar orientations and the response surface method.

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3.9.2

Neighborhood Search Conclusions

We have shown that for head-and-neck cases, quality plans with fewer beams than a standard treatment plan can be obtained if BOO is applied. The simulated annealing and Add/Drop algorithms both regularly obtained quality treatment plans with as few as four beams in only 30 minutes. The use of the flip neighborhood improves the rate of FMO convergence in both algorithms, and even has the ability to improve organ sparing as shown in the simulated annealing results. The simulated annealing and Add/Drop algorithms performed comparably to each other, with neither algorithm indicating a significant benefit over the other. It is possible to incorporate flip neighborhoods into other BOO algorithms that rely on neighborhood searches to yield improved treatment plans in clinically acceptable time frames.

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CHAPTER 4 FRACTIONATION 4.1

Introduction

Typically, head-and-neck treatment plans each contain two target structures, or planning tumor volumes(PTV): PTV1 and PTV2. Let PTV1 be the tumor mass observed from imaging scans, and let PTV2 be PTV1 plus some margin specified by the physician. Rather than deliver an entire treatment plan in one session, a treatment plan is divided into several sessions, called fractions. This is done to take advantage of the fact that normal, healthy cells recover faster from the radiation than cancerous cells. To obtain the treatment plans for the fractions, in practice, a single FMO treatment plan is developed and then divided into the desired number of fractions, usually around 35. This division of a treatment plan is a non-trivial task, as the target voxels must receive 1.8-2.0 Gy of radiation in each fraction. With a single IMRT treatment plan, it is practically impossible to devise a constant dose-per-fraction delivery technique because only a single FMO problem is solved to obtain the treatment plan, which is then simply divided into a number of daily fractions. If a single plan is optimized to deliver doses to multiple target-dose levels, then the dose per fraction delivered to each target must change in the ratio of a given dose level to the maximum dose level. For example, say PTV1 has a prescription dose of 70 Gy, PTV2 has a prescription dose of 50 Gy, and the number of fractions is 35. If a single treatment plan is divided among the 35 fractions, then PTV1 will receive 70/35 = 2.0 Gy in each fraction, but PTV2 will only receive 50/35 = 1.4 Gy, and thus any cancerous cells in PTV2 may not be eradicated by the treatment. Similarly, if only 25 fractions are used in order to ensure that PTV2 receives 2.0 Gy per fraction, then PTV1 receives 70/25 = 2.8 Gy per fraction, well above the desired dose. We propose a new method of approaching the fractionation subproblem wherein an FMO treatment plan is developed for each target structure, rather than developing a

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single treatment plan for all target structures. The individual treatment plans can then be easily divided into optimal fractions. The primal-dual interior point algorithm presented by Aleman et al. [88] is used to solve the FMO and fractionation models to optimality. 4.2

Model Formulation

The fractionation model builds on the FMO model described in Chapter 2. To solve the fractionation problem, we consider developing an individual fluence map solution for each target. For a case with two targets, two plans must be developed: (1) a plan that delivers the prescription dose to PTV1 and PTV2, and (2) a plan that “boosts” the dose received by PTV1 to reach the prescribed dose level. These two fluence maps can then be divided into the appropriate number of fractions easily. For the example of 50 Gy and 70 Gy prescription doses for PTV2 and PTV1, respectively, this would yield 25 fractions of treating both PTV1 and PTV2 to 50/25 = 2.0 Gy, and another 10 treatments of treating just PTV1 to (70 − 50)/10 = 2.0 Gy. For simplicity, we call these individual fluence maps “fractions”, rather than using the term to describe the daily treatments. The development of these fluence maps separately would result in suboptimal solutions. To optimize these fluence map sets simultaneously, we consider each bixel in each fraction as an individual decision variable. As there number of fractions is equal to the number of targets (T ), this results in a fluence map developed for each target. In the single FMO formulation, dose to voxel j in structure s is defined as zjs = PN

i=1

Dij xi , s = 1, . . . , T , and the penalty associated with it as Fs (zjs ). Because the

fractionation model will be concerned with dose-per-fraction as well as cumulative dose, new variables must be defined to express these values.

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Define xfi , f = 1, . . . , T , as the fluence of beamlet i in fraction f . The amount of dose received by a voxel j in structure s in fraction f is defined as f zjs

=

N X

Dij xfi ,

j = 1, . . . , vs , s = 1, . . . , T, f = 1, . . . , T

(4–1)

i=1

Critical structures are thought to be affected by only the cumulative dose received from all treatments, rather than the just the dose in any one particular fraction. This cumulative dose received by a voxel is zjs =

T X N X

Dij xfi ,

j = 1, . . . , vs , s = T + 1, . . . , S

(4–2)

f =1 i=1

Critical structures are penalized in the same manner as in the original FMO model, that is, Fs (zjs ), s = T + 1, . . . , S. Targets require a more complex treatment in the fractionation model. In each fraction, we are primarily concerned with dose received by the targets in that particular fraction. Thus, new variables are needed to express the amount of dose per fraction f received by a voxel (zjs in Equation (4–1)).

Since we must also ensure that the cumulative dose received by each target reaches the prescribed dose, variables to express the cumulative dose received by a voxel are required. Intuitively, this cumulative dose should be the sum of all the doses received in all fractions. If the cumulative dose for targets is defined this way, then over/underdosing in one fraction can result in under/overdosing in another to compensate, which is undesirable. To prevent such a scenario, another new variable called the artificial dose is required (¯ zjs in Equation (4–3)). Rather than simply summing up the dose received in each fraction, we will assume that in the previous fraction, the target voxel received exactly the correct prescription dose for the previous fraction. Thus, no compensating will be necessary. The artificial dose is just the prescription dose from the previous fraction

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(Pf −1 ) plus the dose received in the current fraction: f z¯js = Pf −1 + zjs

j = 1, . . . , vs , s = 1, . . . , T, f = 1, . . . , T

(4–3)

Since each of the target voxels being irradiated in fraction f is treated as target f , the penalty functions for these voxels is T X X

f Ff (¯ zjs )

s=f j∈Vs

Once a target has received its prescription dose, ideally, it should not receive any further dose. As target f is treated in fraction f , for all fractions after f , target f should be treated as normal tissue. Specifically, targets that no longer require dose will be treated as skin, denoted structure S. Therefore, these target voxels, along with actual skin voxels, will be penalized with penalty function FS . The dose received by these target voxels is the prescription dose of the voxel (Ps ) plus the dose received in all subsequent fractions P ` ). This leads to the following penalty functions for voxels penalized as normal ( T`=s+1 zjs tissue in fraction f : f −1 X X

FS

Ps +

s=1 j∈Vs

T X `=s+1

! ` zjs

+

X

FS (zjS )

j∈VS

As with the traditional FMO model, penalty functions are normalized according to the number of voxels in the structure. For critical structures, this normalization factor is still 1/vs since there are always vs voxels being treated as critical structure s. In each fraction, the number of target voxels depends on which targets still need to be treated. Each fluence map set will only “see” the target voxels that are included in its prescription dose level. Thus, define the number of target voxels treated in fluence map f as v¯f =

T X

vs

f = 1, . . . , T

s=f

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The number of voxels treated as skin in each iteration can be expressed by v¯1 − v¯f + vS , where v¯1 − v¯f is the number of target voxels being treated as skin and vS is the number of actual skin/unspecified tissue voxels. Identical to the traditional FMO, the critical structures are normalized and penalized by S−1 X 1 X Fs (zjs ) vs j∈V s=T +1 s

f f Let z be a vector of all zjs , zjs and z¯js variables. The objective function is obtained

by summing the normalized penalty functions: ( " f −1 T X XX 1 FS Ff rac (z) = v¯1 − v¯f + vS s=1 j∈V f =1

Ps +

1 + v¯f

X

s=f j∈Vs

#

! ` zjs

+

`=s+1

s

T X

T X

X

FS (zjS )

j∈VS

) X 1 f Ff (¯ zjs )+ Fs (zjs ) v s s=T +1 j∈V S−1 X

s

The fractionation model is then formulated as minimize Ff rac (z) N X f subject to zjs = Dij xfi

j = 1, . . . , vs , s = 1, . . . , T, f = 1, . . . , T

i=1

zjs =

T X N X

Dij xfi

j = 1, . . . , vs , s = T + 1, . . . , S

f =1 i=1 f z¯js = Pf −1 + zjs

j = 1, . . . , vs , s = 1, . . . , T, f = 1, . . . , T

x≥0 As the objective function is the sum of quadratic functions and the constraints are all linear, the fractionation formulation, just like the basic FMO formulation. 4.3

Results

The fractionation model is tested using the primal-dual interior point algorithm in Aleman et al. [88]. One significant benefit of employing a primal-dual interior point algorithm is that the solution generated is guaranteed to be optimal to within a certain

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tolerance that can be specified by the user. Thirteen head-and-neck cases using five equi-spaced beams are tested. Each test case consists of two targets, PTV1 and PTV2, with prescription dose levels of 70 Gy and 50 Gy, respectively. According the suggestions made on algorithm parameters in Aleman et al. [88], the primal-dual interior point algorithm was implemented with a Single Approximation Hessian and a stopping criteria of a relative duality gap of 0.1%. Although it was also recommend to remove “insignificant” beamlets, these removal of these beamlets actually increases run time in the fractionation model. Thus, insignificant beamlets are left in the fractionation model. 4.3.1

Computational Results

The tests are run in Matlab (MathWorks, Inc.) on a 2.33GHz Intel Core 2 Duo processor with 2GB of RAM. Table 4-1 shows the sizes of each case in terms of the number of decision variables (the number of bixels) and the size of the patient area being treated (the number of voxels). The computation times obtained are display in Table 4-1. On average, the fractionation model was solved in 22.03 seconds. With the same algorithm parameters and weighting parameters, a single FMO treatment plan can be determined in an average of 16.28 seconds, thus there is only a 35% increase in computation time required to develop two FMO plans for the fractionation model. This relatively small increase in time could be accounted for by the fact that the weighting parameters used in the objective function were specifically tuned for the fractionation model. Using parameters specifically tuned to the single-FMO model, the single-FMO model can be solved on average in 9.36 seconds. Compared to this average run time, the FMO model requires 2.4 times as much computation time to develop two models as opposed to one, which is a more intuitive expectation of the interior point method’s performance.

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Table 4-1. Case sizes and run times using identical algorithm and weighting parameters. Single FMO Fractionation Case Bixels Voxels Iterations Time (s) Iterations Time (s) 1 813 85,017 16 8.39 16 19.60 2 1320 189,234 103 82.69 14 55.34 3 935 86,255 24 11.75 11 18.79 4 692 58,636 15 6.87 11 11.47 5 1044 102,262 14 13.16 12 29.70 6 1005 84,369 13 10.31 12 25.58 7 822 71,873 17 9.14 14 18.88 8 802 92,307 59 22.92 14 20.19 9 911 65,541 18 10.84 17 26.12 10 642 66,634 25 7.94 16 12.44 11 279 56,847 29 2.75 14 2.99 12 994 96,105 17 12.30 12 27.13 13 823 72,729 33 12.55 14 18.15 Average 852 86,755 29 16.28 14 22.03 4.3.2

Clinical Results

Because there is no fundamental way of quantifying a treatment plan, DVHs are examined in addition to objective function values to assess the quality of a treatment plan.. The prescription doses used are 70 Gy for PTV1 and 50 Gy for PTV2. These are common prescriptions used in the cancer center at Shands Hospital at the University of Florida. Figures 4-1-4-7 show both dose volume histograms (DVHs) and axial slices for several cases. The DVHs show that in the first fraction, both PTV1 and PTV2 are treated to 50 Gy, and in the second fraction, only PTV1 is treated to an additional 20 Gy. The prescription dose for the fraction is marked by a vertical line. The amount of dose received by each target in each fraction is clinically acceptable. As this study focuses on head-and-neck cases where the most conflict lies in treating the targets while sparing the saliva glands, only DVHs of the saliva glands are shown. All other organs, including skin/unspecified tissue, receive a low enough amount of dose to be spared in the treatment. The sparing criteria for each of the common critical structures in head-and-neck cases are listed in Table 4-2. The critical structures involved in each case

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Table 4-2. Sparing criteria varies for each critical structure Structure Percent (%) ≤ Dose (Gy) brain stem 100 55 eyes 50 30 mandible 100 70 optic chiasm 100 55 optic nerves 100 50 parotid glands 50 30 skin 100 60 spinal cord 100 45 submandibular glands 50 30 vary, depending on their proximity to the tumor, and thus DVHs for some cases do not include all saliva glands. DVHs of the saliva gland doses in Fraction 1 show that the saliva glands receive the majority of dose in the first fraction. Because the cumulative amount of dose received determines whether or not critical structures can be spared, the DVHs for Fraction 2 depict the cumulative dose of these organs. The sparing criteria used for saliva glands is that no more than 50% of the gland can receive more than 30 Gy. This point is marked as a star. For most cases, all of the saliva glands are spared. Figures 4-1-4-7 also show the dose received in each fraction as a colorwash of a slice of the patient. Fraction 1 delivers a homogeneous dose of 50 Gy to both PTV1 and PTV2 while generally avoiding overdosing any of the marked critical structures. In Fraction 2, the dose to PTV1 is boosted by 20 Gy without delivering any unnecessary dose. 4.3.3

Spatial Coefficient Results

The concept of employing spatial information as described in Section 2.4 is also applied to the fractionation model. One set of spatial coefficients is used to obtain both fractions. For the fractionation treatment plans, the spatial coefficients are λ = 1.02, µ = −0.92, β = 0.97 and the minimum coefficient for target voxels is 0.6. Generally, the DVHs for both targets and critical structures using spatial coefficients are similar to those obtained without using spatial coefficients. In fact, in the cases tested,

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Target DVHs: Fraction 1 of 2 PTV1 ∪ PTV2

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Target DVHs: Cumulative dose

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Target DVHs: Fraction 2 of 2

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Figure 4-1. Target DVHs, saliva DVHs and axial slices in Fractions 1 (left) and 2 (right).

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left parotid gland right parotid gland

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there were no instances of either the spatial treatment plans or the non-spatial treatment plans yielding clinically significant changes in the DVHs. The slices show that there is improved homogeneity in the target doses when spatial coefficients are used. The slices also indicate that since the use of spatial coefficients results in the target voxels weighing more heavily than other voxels, the model is more willing to deliver dose to critical structures rather than overdose or underdose the target. This helps provide a uniform dose in the target, and should still be acceptable as the cumulative dose for all critical structures remains within acceptable levels and there are no instances of sacrificing organs that were not already sacrificed in the non-spatial plan. Because more critical structure voxels receive dose in the spatial plans, the dose deposited in the target structures is more spread out, and thus the maximum dose received by the critical structure voxels is less than in the non-spatial plans. This of course means that more voxels are exposed to radiation, but the levels are lower and the amount of radiation still falls within clinically acceptable limits. The resulting improvement in homogeneity is evident for each of the cases, but the effect of the more spread out dose is best illustrated in the second fraction of each case. Figures 4-8–4-14 show the DVHs and slices for some of the tested cases. In particular, Figures 4-9, 4-10, 4-11 and 4-14 demonstrate that the spatial coefficients reduce the amount of dose delivered outside of the targets when compared to their respective non-spatial plans in Figures 4-2, 4-3, 4-4 and 4-7. 4.4


The fractionation model presented allows for the creation of guaranteed optimal fluence maps for each fraction of a patient’s treatment. These fluence maps can be easily divided into the appropriate number of fractions without sacrificing optimality. Using the primal dual interior point method, the fractionation model obtains fluence maps for each target in a clinically feasible amount of time. As expected, the computation time required to generate two fluence maps for a two-target case is more than the time necessary to

111


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Figure 4-8. Target DVHs, saliva DVHs and axial slices in Fractions 1 (left) and 2 (right) using spatial coefficients.

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Figure 4-9. Target DVHs, saliva DVHs and axial slices in Fractions 1 (left) and 2 (right) using spatial coefficients. 113


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Figure 4-10. Target DVHs, saliva DVHs and axial slices in Fractions 1 (left) and 2 (right) using spatial coefficients. 114


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generate a single FMO plan, but the computation times are still acceptable. Further parameter tuning could possibly yield better results. The addition of spatial coefficients in the model allows for improved homogeneity, but does not seem likely to provide additional organ sparing. The improved homogeneity alone is enough to warrant the inclusion of spatial information in the model. The model is sensitive to the changes in the spatial coefficients, so further parameter tuning will have to be performed in small incremental changes. Currently, the model assumes that prior to each fraction, each target voxel has received exactly the prescribed amount of dose up to that point in time. While we have assumed that over/underdose in one fraction should not be compensated by under/overdose in another fraction, it may in fact be advantageous to allow for some degree of compensation. The fractionation formulation proposed affords enough flexibility to model such a scenario. For example, say a physician would like to allow underdose in target s in previous fractions to be compensated by up to ξ Gy of overdose in the current fraction. Then, for target structure s, the Ps term in the objective function would be replaced by the expression max{zjs , Ps − ξ}. As this type of discontinuity already exists in the model, the structure of the model would not be altered by making this modification. Other future research possibilities include further parameter testing to employ the model on other cancer site treatments, for example, lung and prostate cancers.

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CHAPTER 5 A MONTE CARLO METHOD FOR MODELING DOSE DEPOSITION 5.1

Introduction

The FMO problem relies on the calculation of the amount of total radiation dose received in each voxel. The dose in a voxel is determined by the paths the photons in the radiation beams follow through the patient. Some photons may collide with particles inside the patient and scatter in any direction, while others may collide with particles and be absorbed. Still other photons may pass entirely through the patient with no collisions. Due to the unpredictable nature of the radiation beam inside the patient, the dose received in a voxel can only be accurately obtained through Monte Carlo simulations. A simple linear relationship is assumed between total dose and beamlet fluences and is commonly accepted as a satisfactory dose approximation in IMRT optimization. Errors of as much as 30% have been reported for photon beams near tissue inhomogeneities (Ma et al. [5]). For IMRT optimization, particularly with advent of image-guided IMRT (IGIMRT), or 4D IMRT, the FMO problem must be solved extremely quickly to create real-time treatment plans. Thus, the speed of the FMO problem is paramount. While Monte Carlo simulation may provide the most accurate measure of dose, the lengthy computation time renders the method impractical for clinical use. We propose a Monte Carlo method that performs a limited number of histories to obtain a noisy approximation of the dose distribution of each beamlet to which a smoothing function can be applied in order to determine an accurate dose distribution. The anticipation is that few histories will be required, and that this approach can be clinically feasible. Recently, a similar approach has been taken by Jele´ n and Alber [89] and Jele´ n et al. [90] with good results. Jele´ n et al. [90] acknowledge that there is some loss of accuracy at the beam’s edge due to a lack of lateral density correction and the effects arising from MLC systems, for example, tongue-and-groove and inter-leaf scatter. Jele´ n and Alber

120

[89] pursue the issue of density scaling, but the MLC effects have not yet been addressed. Section 5.6 proposes some possible methods of accounting for such MLC effects. 5.2

Monte Carlo Engine

The “Dose Planning Method” (DPM) (Sempau et al. [91]) program will be used to perform the Monte Carlo simulations. DPM is designed to simulate the transport of photons in radiotherapy class problems. DPM is primarily based on the public domain code PENELOPE (Baró et al. [92], Sempau et al. [93]). This study focuses on modeling a finite sized pencil beam emanating from a 6MV linear accelerator. A finite sized pencil beam is a beam of finite sized that is parallel to the point source of radiation. To determine a reasonably accurate measure of the dose of a single beamlet in a given tissue, approximately one billion histories are run in DPM. As fewer histories are run, the inaccuracies of the dose resulting from the Monte Carlo experiment grow. Figure 5-4 shows how the noise in the depth-dose curve of the beamlet becomes increasingly pronounced in relation to the number of histories. As shown by Table 5-1, the amount of time required to run each experiment is approximately linear in the number of histories recorded. Thus, it is impractical to run the number of histories necessary for acceptable accuracy. 5.3

Dose Distribution of a Beamlet

The accuracy of a treatment plan is contingent upon the accuracy of the calculated dose deposited by each beamlet in the plan. Because the particles in a beamlet scatter in three dimensional space, multiple dose distributions must be considered to satisfactorily model the beamlet’s affect on the patient’s tissue. These distributions arise from the amount of radiation the beamlet deposits as a function of depth (the depth-dose curve), and from the amount of radiation radiating outward from the center of the beamlet (the lateral penumbra).

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5.3.1

Depth-Dose Curve

The depth-dose curve represents the radiation intensity deposited by the beamlet in the tissue through which it passes as function of depth. Figure 5-1 shows the dose distribution of a single 6MV beamlet in various tissues obtained from the DPM simulations. The dose distribution of a beamlet in water is empirically known, and the results from the DPM simulation in water can be easily verified to be correct. Muscle, which has nearly identical density as water (the densities of muscle and water are 1.04g/cm3 and 1.00g/cm3 , respectively), has nearly the same depth-dose distribution as water. As expected, a beamlet passing through lung tissue, which is significantly less dense than water, does not lose its intensity as quickly as it travels through the less dense tissue. Lastly, a simulation with inhomogeneous tissue is considered. A simulation of muscle with a 10-cm thick layer of lung located at a depth of 10cm shows a dose distribution that when the beamlet reaches the less-dense segment of lung, its depth-dose curve becomes less steep, indicating that less radiation intensity is lost through the lung than through the muscle. Once the deeper layer of muscle is reached, the steepness of the depth-dose curve increases again. Depth dose curve of beamlet in various tissues after 1B histories 1.1 water muscle lung muscle−lung−muscle

1 0.9

relative dose (%)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

5

10

15 depth (cm)

20

25

Figure 5-1. Dose distribution of a single beamlet in various tissues.

122

30

Although it may seem unintuitive that the depth-dose curve increases at shallow depths, this behavior is called the build-up curve, and is explained by the likelihood of electrons scattering out of the tissue and into air at shallow depths. Because the density of air is extremely small, an electron that reaches air is likely to travel very far away from the tissue, and therefore unlikely to return to the tissue and deposit radiation dose. Once the depth increases passes a certain point, the electrons cannot leave the tissue and the amount of dose received in the tissue increases. Once that point is reached, the amount of radiation delivered by the beamlet decreases monotonically in depth, as would be expected. 5.3.2

Lateral Penumbra

In addition to the dose distribution occuring as the beamlet penetrates the tissue, there is a dose distribution spreading away from the beamlet. Just as light emanating from a flashlight in a dark room does not have a discrete boundary between light and dark, the radiation delivered by a beamlet also does not have a discrete boundary between what is and is not irradiated. With a circular flashlight beam shown onto a flat surface, it is apparent from the distribution of the illuminated portion of the surface that some of the light is diffused into the surrounding darkness as a result of scatter. If the distribution of light in the circular projection of the flashlight beam is plotted, a bell-shaped curve describes the brightest point in the center of the illuminated disc decreasing in brightness as the edge of the illuminated disc is approached, eventually reaching complete darkness. This behavior is parallel to the behavior of a beamlet passing through any medium. From The Physics of Radiation Therapy [94], the penumbra of a beam is the region at the edge of a radiation beam, over which the dose rate changes rapidly as a function of distance from the beam axis. Hence, the distribution of radiation dose originating from the beamlet described above is called the lateral penumbra. Figure 5-2 shows the colorwash of dose distribution consistituting the lateral penumbra, while Figure 5-3 shows the dose

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Lateral penumbra of a finite sized pencil beam

−4

x 10 11 10 9 8 7 6

0 5 4 3 2 1 0 distance from beam

Figure 5-2. Colorwash of the lateral penumbra of a finite sized pencil beam distribution of the lateral penumbra at a fixed depth in one dimension obtained from one billion Monte Carlo histories of a 5-cm finite sized pencil beam in water . 5.4

Methodology to Model a Beamlet

Modeling the dose distribution of a beamlet is relatively straightforward for a beamlet in a single medium. The difficulty arises when multiple mediums are traversed by the beamlet because the varying densities affect the particle scattering of the beam, thus affecting both the depth-dose curve and the lateral penumbra. As previously stated, errors of as much as 30% have been reported for photon beams near tissue inhomogeneities (Ma et al. [5]). Because there are numerous inhomogeneities in most cancer treatment sites, these inhomogeneities are of particular interest. The beamlet’s behaviour at the boundary of different tissue types cannot be determined as easily, and thus requires Monte Carlo simulation. In designing an IMRT treatment plan for a patient, there can be more than a dozen different structures (tissue types) with complicated boundary geometries.

124

−3

1.2

x 10

Lateral penumbra of 5−cm finite size pencil beam

1

dose (Gy)

0.8

0.6

0.4

0.2

0

0

2

4

6 distance (cm)

8

10

12

Figure 5-3. Plot of the lateral penumbra of a finite sized pencil beam Knowledge of a beamlet’s behaviour given certain tissue inhomogeneities can be very useful in accurately determining dose in a voxel. 5.4.1

Modeling the Depth-Dose Curve

In the section, we analyze the behavior of the depth-dose curve under both single tissue and multiple tissue scenarios. The goal of the analyzation is to determine the minimum number of Monte Carlo histories required to obtain a reasonably accurate approximating function of the dose deposited at each depth in the tissue. For both the instances of only a single medium and multiple mediums, this is done by fitting the depth-dose curve from Monte Carlo experiments with varying numbers of histories to high-degree polynomial functions. The polynomial fits are then compared to the polynomial fit of a very accurate measure of the depth-dose curve obtained from an number of Monte Carlo histories accepted to be satisfactorily accurate. The number of histories recorded in the Monte Carlo simulation can have a drastic effect on the accuracy of the data collected. For example, Figure 5-4 demonstrates the vast

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Depth dose curve of beamlet in water 1.4 1B histories 100M histories 10M histories 1M histories

1.2

relative dose (%)

1

0.8

0.6

0.4

0.2

0

0

5

10

15 depth (cm)

20

25

30

Figure 5-4. Observed depth-dose curve in water for several histories. variation observed in the depth-dose curve of a beamlet in water for histories ranging from one million to one billion. It is hoped that after a certain number of histories, the function approximation of the data will closely follow the function approximation of very accurate data obtained from a large number of histories. For a beamlet in both homogeneous and heterogeneous tissue, the depth-dose curve can be modeled using a polynomial function of order k. Although the depth-dose curve may exhibit changes in concavity in the presence of tissue inhomogeneity, a high degree polynomial will capture the curve’s behavior. The variation of a k-degree polynomial fitted to n-history Monte Carlo data is measured by

0

vk,n,n0 = d(n ) − p(k,n) , 2

0

where d(n ) is the actual observed depth-dose curve from n0 Monte Carlo histories and p(k,n) is the vector of approximated depth-dose values obtained from a polynomial fit of

126

Polynomial fits compared to 1B−history data in water 1.1 1B histories: k=27, var=0.050278 100M histories: k=23, var=0.080158 10M histories: k=28, var=0.21877 1M histories: k=24, var=0.54071

1 0.9

relative dose (%)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

5

10

15 depth (cm)

20

25

30

Figure 5-5. Polynomial fits of several histories compared to the observed 1B-history depth-dose curve in water. degree k to data obtained from n Monte Carlo histories. It is desirable to have that n0 > n to assess the quality of the polynomial fit compared to more accurate data. In this study, the accuracy of the polynomial obtained is judged by its variation from the observed data from a very large number of Monte Carlo histories, that is, n0 >> n in the calculation of vk,n . Figure 5-5 shows that for the illustrated number of histories, the polynomial fit from 100 million histories closely resembles not only the polynomial fit from one billion histories, but also the actual data collected from one billion histories. The polynomial fit to one million histories is clearly an unsatisfactory approximation to the data collected from one billion histories. For several numbers of Monte Carlo histories, the best approximating polynomial ¯ is found, that is, k ∗ = arg mink∈[k,k] function with degree in the range [k, k] ¯ {vk,n }. Several degrees are tested because the degree of the polynomial can significantly affect the quality of the fit, even for polynomials that are only one degree apart. Figure 5-6 illustrates the amount of variation observed in the polynomial approximation as a function of the degree

127

Variation of polynomial fit (v k,n,1e9 ) as function of degree (k) 0.8 0.7

variation (v

k,n,1e9

)

0.6 0.5 0.4 0.3 0.2 0.1 0

5

10

15

20 25 30 35 degree of polynomial (k)

40

45

50

Figure 5-6. Variation of polynomial fit as function of degree. of the polynomial for polynomials fitted to the depth-dose curve of a beamlet in water obtained from 1 billion histories. 5.4.2

Modeling the Lateral Penumbra

In the section, we analyze the behavior of the lateral penumbra under both single tissue and multiple tissue scenarios. The lateral penumbra of a beam is a bell-shaped curve that can be approximated as the sum of error function pairs. The error function, erf(x), is twice the integral of the Gaussian distribution with mean 0 and variance of 1/2: 2 erf(x) = √ π

Z

x

2

e−t dt.

0

Figure 5-7A demonstrates a sample error function. While a single side of the lateral penumbra of a beamlet resembles an error function, a closer approximation to a single side of the lateral penumbra is represented as the average of two error functions given by a x + x0 x − x0 erf − erf , 2 σ σ

128

Sample error function: erf(x)

Sample error function pair for lateral penumbra

1 1 0.8 0.6

0.8

0.4 0.2

0.6

0 −0.2

0.4

−0.4 −0.6

0.2

−0.8 −1 −3

−2

−1

0

1

2

0

3

0

A

1

2

3

4

5

6

B

Figure 5-7. An error function and an error function pair. A) Error function. B) Error function pair. where a is the amplitude, x0 is the offset and σ is the variation of the two error functions. The expression is divided by 2 to take the average of the error function pair. An example of an error function pair is given in Figure ??B. Because the lateral penumbra of a beamlet resembles an error function on both the left- and right-hand sides of the beam center, the lateral penumbra L(x) is represented as the sum of the average of N error function pairs, given by N X x + x 0i x − x 0i ai erf − erf , L(x) = 2 σi σi i=1 where ai is the amplitude, x0i is the offset and σi is the variation of error function pair i, i = 1, . . . , N . To determine the parameters ai , x0i and σi for each of the N error function pairs, a Levenberg-Marquardt quasi-Newton minimization method is employed. This method takes as input N and an initial guess of the parameters and returns a locally optimal solution to the problem of minimizing the variation between the real data and the sum of the error function pairs. At a given depth in the tissue, the amplitude of the error function is determined by the value of the depth-dose curve at that depth. Thus, for each tissue type, it is only

129

Lateral penumbra of 5−cm finite sized pencil beam in water 1B histories 100M histories 10M histories 1M histories

1

relative dose (%)

0.8

0.6

0.4

0.2

0

0

2

4

6 distance (cm)

8

10

12

Figure 5-8. Lateral penumbra for several numbers of Monte Carlo histories. necessary to model a single lateral penumbra, and then that model can be extended to all depths simply by manipulating the amplitude according to the depth-dose curve. Figure 5-3 shows the lateral penumbra of a 5-cm finite sized beamlet at a fixed depth in water for a number of Monte Carlo histories deemed to yield a satisfactorily accurate representation of the dose deposited in the tissue. Using the method described above, the lateral penumbra was modeled to yield the approximation to the observed data collected for the various Monte Carlo histories shown in Figure 5-8. In a similar fashion to the method for modeling the depth-dose curve, the method for modeling the lateral penumbra consists of fitting the sum of error function pairs to the lateral penumbra data. The quality of these fits is judged by their variation from the observed data for a sufficiently large number of Monte Carlo histories to obtain accurate dose information.

130

Error function pair fits compared to 1B−history data in water 1.2

1B histories: var=0.070667 100M histories: var=0.075414 10M histories: var=0.14511 1M histories: var=1.1829

relative dose (%)

1

0.8

0.6

0.4

0.2

0

0

2

4

6 distance (cm)

8

10

12

Figure 5-9. Error function fits of several histories compared to the observed 1B-history lateral penumbra of a beamlet in water. Just in as the method for determining the quality of the depth-dose curve approximation, the variation of the error function fit from the actual lateral penumbra is calculated as νn,n0

(n0 ) ˆ (n,N ) = L − L

, 2

0

where L(n ) is the observed lateral penumbra data from a simulation of n0 histories, and ˆ (n,N ) is the approximated lateral penumbra obtained from the parameters fitted to the L expression LN (x). It is desirable to have that n0 > n. Figure 5-9 displays the error function pair fits obtained from the Levenburg-Marquardt method, as well as the variation of the fits from the observed data from one billion histories. The variation is measured in the same manner as described in Section 5.4.1. It is anticipated that although the lateral penumbra exhibits different dose distributions in materials of different densities, the distribution will only show a fundamental change in shape if the beam simultaneously hits multiple tissues of varying densities. In such a situation, the penumbra, which is taken to be symmetric about the center of the beam in

131

Table 5-1. Computation times in n Water 1e9 222.184 100e6 20.543 10e6 2.210 1e6 0.244

minutes of Monte Carlo simulations Muscle Lung Muscle-Lung-Muscle 211.887 111.318 186.894 21.256 11.239 18.701 2.234 1.269 1.986 0.339 0.233 0.309

homogeneous tissue, will no longer be symmetric. To model the lateral penumbra under inhomogeneous material, a sum of error function pairs can still be employed, though it may be necessary to increase the number of error function pairs required. The difficulty will lie in correctly determining when the addition of additional error function pair will be needed. A possible measure could be the variation between the lateral penumbra approximation and the observed data. 5.5

Results

The homogeneous tissues tested are water, muscle and lung, and the heterogeneous material tested consists of muscle and lung. Each scenario is considered to have a depth of 30cm. The voxel sizes are 5mm × 5mm × 5mm, and a 5-cm finite sized pencil beam is considered. For each simulation, tests were run with 1 billion, 100 million, 10 million and 1 million Monte Carlo histories in DPM on a Mac OS X 10.4.6 machine with dual 2.3GHz PowerPC G5 processors and 8GB of RAM. Due to time constraints, the muscle tests are run to a maximum of 100 million iterations, and all comparisons to the fit quality are made to this 100-million-history data instead of the 1-billion-history data used for the other simulations. As can be seen from the computation times in Table 5-1, the run time of DPM is approximately linear in the number of histories. Altough a larger number of Monte Carlo histories yields improved accuracy, the maximum number of histories considered is one billion because of time limitations and the satisfactory accuracy of the 1-billion-history runs.

132

For each of the tested tissue types, the depth-dose curves and lateral penumbras were modeled using the methods described in Section 5.4. For the polynomial fits of the depth-dose curve, the values k and k¯ are chosen as 10 and 45, respectively. By choosing the polynomial approximation over such a large range of degree values, an acceptably accurate fit is likely to be found. For the lateral penumbra, N was chosen as 4 because in addition to the obvious need for two error functions to model the sides of the lateral penumbra, an additional error function is needed to model each tail with reasonable accuracy. For example, the four error functions used to model the lateral penumbra of a beamlet in water (Figure 5-9) are shown separately in Figure 5-10. The computation times required to obtain each of the function approximations are displayed in Table 5-2. The initial parameters ai , x0i and σi for each error function pair i, i = 1, . . . , N , used to approximate the lateral penumbra are obtained by the following method. Of the four error function pairs considered, two of the error functions—I = {1, 2}—are used to model the steep sides of the lateral penumbra, and the other two error functions—I¯ = {3, 4}—are used to model the tails of the dose distribution. At a given depth z, the amplitude ai is    d(z) i∈I ai =  ¯  d(z)/50 i ∈ I, where d(z) represents the value of the depth-dose curve approximation at a depth z. The expression for the amplitude when i ∈ I¯ was obtained by experimenting with several different fractions of d(z). The σ value of the error functions determines the shape of the error function curve. As σ increases, the curve becomes increasingly spread out. Thus, it is desirable to have a small σi value for i ∈ I since the error function in I only need to model the sides of the lateral penumbra, and a larger σi value for i ∈ I¯ since the error function in I¯ need to model the elongated tails of the lateral penumbra. For the tissues tested, the σi values

133

Table 5-2. Computation times in seconds of approximating function fits to the dose distribution. The polynomial fits to the depth dose curve are represented by D.D., and the error function fits to the lateral penumbra are represented by Lat.Pen. n 1e9 100e6 10e6 1e6

used are

Water D.D. Lat.Pen. 0.078 2.640 0.078 1.172 0.110 3.454 0.094 1.407

Muscle D.D. Lat.Pen. 0.078 2.422 0.078 2.625 0.109 1.390 0.094 1.172

D.D. 0.094 0.828 2.609 1.063

Lung Lat.Pen.. 1.062 0.906 2.594 0.953

Muscle-Lung-Muscle D.D. Lat.Pen. 0.078 n/a 0.109 n/a 0.093 n/a 0.078 n/a

   0.4 i ∈ I σi =  ¯  0.8 i ∈ I,

These values were obtained through experimentation. For the 5-cm finite sized pencil beams used in this experiment, the offsets x0i were empirically set at values of 8.5, -3.5, 11 and -1 for i = 1, . . . , N , respectively. A method of identifying the locations of these offsets based on the Monte Carlo data can be developed by basing the offsets on the slope of the observed data, and is planned for future research. The results for the fits of both the depth-dose curve and the lateral penumbra of a beamlet in water are shown in the examples in Section 5.4. Figures 5-11-5-12 show the results of the fits for the muscle and lung tissues. From the computational results, it is clear that the time to obtain fits to the Monte Carlo data is insignificant compared with the amount of time required to run the Monte Carlo histories, even for as few as 1 million histories. To test the model in the presence of tissue inhomogeneity, a 10cm-thick layer of lung between two 10cm-thick layers of muscle is considered. As expected, for the first 10cm, the depth-dose curve of the muscle-lung-muscle case is identical to that of the muscle depth-dose curve. Once the beamlet reaches the significantly less dense layer of lung (lung has a density of 0.30g/cm3 ), a predominant change in the depth-dose curve is evident (Figure 5-1). Once the layer of lung is reached, the rate of decrease in the amount of dose deposited in the tissue decreases, that is, less radiation intensity is lost as the beamlet

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Error functions for 5−cm finite sized pencil beam in water 0.8 erf pair 1 erf pair 2 erf pair 3 erf pair 4

0.6

relative dose (%)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

2

4

6 distance (cm)

8

10

12

Figure 5-10. Error function pairs summed to approximate a beamlet in water.

Depth dose curve of beamlet in muscle

Polynomial fits compared to 1B−history data in muscle

1.4

1.4 1B histories 100M histories 10M histories 1M histories

1.2

1 relative dose (%)

relative dose (%)

1

0.8

0.6

0.8

0.6

0.4

0.4

0.2

0.2

0

1B histories: k=27, var=0.046123 100M histories: k=22, var=0.075212 10M histories: k=22, var=0.20284 1M histories: k=24, var=0.69751

1.2

0

5

10

15 depth (cm)

20

25

0

30

A

0

5

10

15 depth (cm)

20

25

30

B

Figure 5-11. Depth-dose curves in muscle tissue. A) Monte Carlo histories. B) Polynomial fits.

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Lateral penumbra of 5−cm finite sized pencil beam in muscle

Error functions for 5−cm finite sized pencil beam in muscle

0.9


0.8

0.7

0.6

relative dose (%)

relative dose (%)

0.7

0.5 0.4

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

2

4

6 distance (cm)

8

10


0.8

0

12

0

2

4

A

6 distance (cm)

8

10

12

B

Figure 5-12. Lateral penumbra curves in muscle tissue. A) Monte Carlo histories. B) Error function fits.

Depth dose curve of beamlet in lung

Polynomial fits compared to 1B−history data in lung

1.5



1.2

relative dose (%)

relative dose (%)

1 1

0.8

0.6

0.5 0.4

0.2

0

0

5

10

15 depth (cm)

20

25

0

30

A

0

5

10

15 depth (cm)

20

25

30

B

Figure 5-13. Depth-dose curves in lung tissue. A) Monte Carlo histories. B) Polynomial fits.

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Lateral penumbra of 5−cm finite sized pencil beam in lung

Error function pair fits compared to 1B−history data in lung 1.2

1B histories 100M histories 10M histories 1M histories

1.2

1

relative dose (%)

relative dose (%)

1

0.8

0.6

0.8

0.6

0.4

0.4

0.2

0.2

0

0

2

4

6 distance (cm)

8

10


0

12

A

0

2

4

6 distance (cm)

8

10

12

B

Figure 5-14. Lateral penumbra curves in lung tissue. A) Monte Carlo histories. B) Error function fits. passes through the lung tissue. When the beamlet reaches the second layer of muscle, this rate increases again. The same approach used to model the depth-dose curve in a single tissue continues to work well in multiple tissue. Figures 5-15A and 5-15B illustrate the ability of a polynomial to approximate the depth-dose curve in inhomogeneous tissue. Because testing the beamlet in a scenario where it could hit multiple tissues simultaneously is reserved for future research, results for modeling the lateral penumbra in the multiple-tissue scenario tested are identical to those for the single-tissue scenario. The lateral penumbra at a given depth in a certain tissue can be modeled by using the dose from the depth-dose curve at the given depth as the amplitude of the lateral penumbra. The dose distribution in the lateral penumbra can then be modeled according to the same error function pairs used in modeling the lateral penumbra in a single-tissue scenario of the same medium. Figure 5-16 illustrates the variations of the fits used to approximate the depth-dose and lateral penumbra distributions of a beamlet in water as a function of the number of histories. From this data, it is very clear that the accuracy of the beamlet model is directly correlated with the number of Monte Carlo histories. It is interesting that there is not a significant improvement in the beamlet model accuracy from 100 million to 1

137

Depth dose curve of beamlet in muscle−lung−muscle

Polynomial fits compared to 1B−history data in muscle−lung−muscle

1.6


1.4

1

1.2

0.9 relative dose (%)

relative dose (%)


1.1

1

0.8

0.8 0.7 0.6 0.5

0.6

0.4 0.4 0.3 0.2

0

5

10

15 depth (cm)

20

25

0.2

30

0

5

10

15 depth (cm)

A

20

25

30

B

Figure 5-15. Depth-dose curves in heterogeneous muscle and lung tissue. A) Monte Carlo histories. B) Polynomial fits. Table 5-3. Variation of fits to several numbers of histories with n0 = 1 billion. n 1e9 100e6 10e6 1e6

Water vk∗ ,n,n0 νn,n0 0.050 0.071 0.080 0.075 0.219 0.145 0.541 1.183

Muscle vk∗ ,n,n0 νn,n0 0.046 0.105 0.075 0.118 0.203 0.124 0.698 0.209

Lung vk∗ ,n,n0 νn,n0 0.057 0.097 0.110 0.103 0.330 0.125 1.055 0.129

Muscle-Lung-Muscle vk∗ ,n,n0 νn,n0 0.052 n/a 0.101 n/a 0.213 n/a 0.881 n/a

billion histories, and computing 100 million histories requires approximately one tenth of the amount of time as computing 1 billion histories. Depending on the composition of the tissue, it may be reasonably accurate to only require 10 million histories, particularly in the depth-dose curve approximation. 5.6


In conclusion, the Monte Carlo approach presented is employed to model the dose distribution of a beamlet using a limited number of histories. Using the polynomial and error function pair fitting techniques described, dose distributions with satisfactory accuracy can be obtained using at least a factor of 10 fewer Monte Carlo histories than would otherwise be required. This can greatly decrease the amount of time required to obtain dose data for beamlets in the FMO problem of IMRT treatment planning without sacrificing accuracy.

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Figure 5-16. Variations of the fits used to approximate the depth-dose and lateral penumbra distributions as function of the number of histories. Variation of approximations as a function of number of histories 1.4 Depth−dose: water Lateral penumbra: water Depth−dose: muscle Lateral penumbra: muscle Depth−dose: lung Lateral penumbra: lung Depth−dose: muscle−lung−muscle

1.2

variation

1

0.8

0.6

0.4

0.2

0 6 10

7

8

10

10

9

10

number of histories

For future testing, more tests on the number of Monte Carlo histories needed will be run as well, particularly with histories in the range of 10-100 million. More tests of varying tissues, both homogeneous and heterogeneous, will be run to determine a smaller range of degrees to be evaluated for the polynomial fit to the depth-dose curve. An automated method of determining a quality set of initial parameters to model the lateral penumbra will also be developed. Lastly, the scenario where a beamlet hits multiple tissues simultaneously will be tested using our model for approximating the lateral penumbra. Jele´ n and Alber [89] and Jele´ n et al. [90] have demonstrated that a beamlet can be modeled very effectively using an approach based on the one described here. This approach was improved upon by scaling the modeling parameters according to tissue density in Jele´ n and Alber [89]. Despite the sophistication of the density scaling method employed, the model loses accuracy in the penumbra regions and at the edge of tissue heterogeneities. This study also used a Levenberg-Marquardt algorithm to determine the

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modeling parameters, and although the details of the implementation are not provided, it is possible that with an improved initial guess or damping parameter, the algorithm could converge to better modeling parameters, thus providing improved prediction of beamlet behavior at the penumbra. To further improve upon their work, the effects of the MLC must be considered. One method of accounting for these effects could be to model the dose deposition of an entire aperture rather than just the dose deposition of a single beamlet. As the number and shape of apertures required to deliver an FMO-based IMRT optimization are unknown, this method would be most practical if an aperture modulation approach—where aperture fluences from a pre-defined set of apertures are chosen, instead of fluences from individual beamlets—is employed instead of an FMO approach, as the number and shape of the apertures in consideration are predetermined.

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REFERENCES [1] American Cancer Society. Cancer Facts and Figures Report. 2006. [2] Murphy GP, Lawrence WL, Lenlard RE, eds. American Cancer Society Textbook on Clinical Oncology. The American Cancer Society, 1995. [3] Perez CA, Brady LW. Principles and Practice of Radiotherapy. Lippincott-Raven, 3 edn., 1998. [4] Steel GG. Basic Clinical Radiobiology for Radiation Oncologists. Edward Arnold Publishers, 1994. [5] Ma CM, Mok E, Kapur A, Findley D, Brain S, Boyer AL. Clinical implementation of a monte carlo treatment planning system. Medical Physics 1999;26:2133–43. [6] Bortfeld T. Optimized planning using physical objectives and constraints. Semin Radiat Oncol 1999;9:20–34. [7] Alber M, Nusslin F. An objective function for radiation treatment optimization based on local biological measures. Phys Med Biol 1999;44:479–493. [8] Jones LC, Hoban PW. Treatment plan comparison using equivalent uniform biologically effective dose (eubed). Phys Med Biol 2000;45:159–170. [9] Kallman P, Lind BK, Brahme A. An algorithm for maximizing the probability of complication–free tumor–control in radiation-therapy. Phys Med Biol 1992; 37:871–890. [10] Mavroidis P, Lind BK, Brahme A. Biologically effective uniform dose for specification, report and comparison of dose response relations and treatment plans. Phys Med Biol 2001;46:2607–2630. [11] Niemierko A. Reporting and analyzing dose distributions: a concept of equivalent uniform dose. Medical Physics 1997;24:103–110. [12] Niemierko A, Urie M, Goitein M. Optimization of 3d radiation-therapy with both physical and biological end-points and constraints. Int J Radiat Oncol Biol Phys 1992;23:99–108. [13] Wu QW, Djajaputra D, Wu Y, Zhou JN, Liu HH, Mohan R. Intensity-modulated radiotherapy optimization with geud-guided dose-volume objectives. Phys Med Biol 2003;48:279–291. [14] Wu QW, Mohan R, Niemierko A, Schmidt-Ullrich R. Optimization of intensity-modulated radiotherapy plans based on the equivalent uniform dose. Int J Radiat Oncol Biol Phys 2002;52:224–235.

141

[15] Hamacher HW, K¨ ufer KH. Inverse radiation therapy planning a multiple objective optimization approach. Discrete Applied Mathematics 2002;118:145–161. [16] Bednarz G, Michalski D, Houser C, Huq MS, Xiao Y, Anne PR, Galvin JM. The use of mixed-integer programming for inverse treatment planning with pre-defined field segments. Phys Med Biol 2002;47:2235–2245. [17] Ferris MC, Meyer RR, D’Souza W. Radiation treatment planning: Mixed integer programming formulations and approaches. In G Appa, L Pitsoulis, HP Williams, eds., Handbook on Modelling for Discrete Optimization. Springer-Verlag, New York, NY, 2006;317–340. [18] Langer M, Brown R, Urie M, Leong J, Stracher M, Shapiro J. Large-scale optimization of beam weights under dose-volume restrictions. Int J Radiat Oncol Biol Phys 1990;18:887–893. [19] Langer M, Morrill S, Brown R, , Lee O, Lane R. A comparison of mixed integer programming and fast simulated annealing for optimizing beam weights in radiation therapy. Medical Physics 1996;23:957–964. [20] Lee EK, Fox T, Crocker I. Simultaneous beam geometry and intensity map optimization in intensity-modulated radiation therapy treatment planning. Annals of Operations Research 2003;119:165–181. [21] Lee EK, Fox T, Crocker I. Integer programming applied to intensity-modulated radiation therapy treatment planning. Int J Radiat Oncol Biol Phys 2006;64:301–320. [22] Shepard DM, Ferris MC, Olivera GH, Mackie TR. Optimizing the delivery of radiation therapy to cancer patients. SIAM Review 1999;41:721–744. [23] Romeijn HE, Ahuja RK, Dempsey JF, Kumar A, Li JG. A novel linear programming approach to fluence map optimization for intensity modulated radiation therapy treatment planning. Phys Med Biol 2003;38:3521–3542. [24] Romeijn HE, Ahuja RK, Dempsey JF, Kumar A, Li JG. A column generation approach to radiation therapy treatment planning using aperature modulation. SIAM Journal of Optimization 2005;15:838–862. [25] Romeijn HE, Dempsey JF, Li JG. A unifying framework for multi-criteria fluence map optimization models. Phys Med Biol 2004;49:1991–2013. [26] Romeijn HE, Ahuja RK, Dempsey JF, Kumar A. A new linear programming approach to radiation therapy treatment planning problems. Operations Research 2006;54:201–216. [27] Das SK, Marks LB. Selection of coplanar or noncoplanar beams using three-dimensional optimization based on maximum beam separation and minimized

142

nontarget irradiation. Int J Radiat Oncol Biol Phys 1997;38:643–655. [28] Haas OC, Burnham KJ, Mills J. Optimization of beam orientation in radiotherapy using planar geometry. Phys Med Biol 1998;43:2179–2193. [29] Schreibmann E, Lahanas M, Xing L, Baltas D. Multiobjective evolutionary optimization of the number of beams, their orientations and weights for intensity-modulated radiation therapy. Phys Med Biol 2004;49:747–770. [30] Chao KSC, Blanco AI, Dempsey JF. A conceptual model integrating spatial information to assess target volume coverage for IMRT treatment planning. Int J Radiat Oncol Biol Phys 2003;56:1438–1449. [31] Nocedal J, Wright SJ. Numerical Optimization. Springer-Verlag, 1999. [32] Ezzell GA. Genetic and geometric optimization of three-dimensional radiation therapy treatment planning. Medical Physics 1996;23:293–305. [33] Li Y, Yao J, Yao D. Automatic beam angle selection in IMRT planning using genetic algorithm. Phys Med Biol 2004;49:1915–1932. [34] Li Y, Yao J, Yao D, Chen W. A particle swarm optimization algorithm for beam angle selection in intensity-modulated radiotherapy planning. Phys Med Biol 2005; 50:3491–3514. [35] Bortfeld T, Schlegel W. Optimization of beam orientations in radiation therapy: some theoretical considerations. Phys Med Biol 1993;38:291–304. [36] Djajaputra D, Wu Q, Wu Y, Mohan R. Algorithm and performance of a clinical IMRT beam-angle optimization system. Phys Med Biol 2003;48:3191–3212. [37] Lu HM, Kooy HM, Leber ZH, Ledoux RJ. Optimized beam planning for linear accelerator-based stereotactic radiosurgery. Int J Radiat Oncol Biol Phys 1997; 39:1183–1189. [38] Pugachev A, Xing L. Incorporating prior knowledge into beam orientation optimization in IMRT. Int J Radiat Oncol Biol Phys 2002;54:1565–1574. [39] Rowbottom CG, Oldham M, Webb S. Constrained customization of non-coplanar beam orientations in radiotherapy of brain tumours. Phys Med Biol 1999a;44:383–399. [40] Stein J, Mohan R, Wang XH, Bortfeld T, Wu Q, Preiser K, Ling CC, Schlegel W. Number and orientations of beams in intensity-modulated radiation treatments. Medical Physics 1997;24:149–160. [41] Söderstrom S, Brahme A. Selection of suitable beam orientations in radiation therapy using entropy and fourier transform measures. Phys Med Biol 1992;37:911–924.

143

[42] Söderstrom S, Brahme A. Which is the most suitable number of photon beam portals in coplanar radiation therapy? Int J Radiat Oncol Biol Phys 1995;33:151–59. [43] Rowbottom CG, Webb S, Oldham M. Beam-orientation customization using an artificial neural network. Phys Med Biol 1999b;44:2251–2262. [44] Gokhale P, Hussein EM, Kulkarni N. The use of beams eye view volumetrics in the selection of non-coplanar radiation portals. Medical Physics 1994;23:153–163. [45] Meedt G, Alber M, N¨ usslin F. Non-coplanar beam direction optimization for intensity-modulated radiotherapy. Phys Med Biol 2003;48:2999–3019. [46] Chen GT, Spelbring DR, Pelizzari CA, Balter JM, Myrianthopoulos LC, Vijayakumar S, Halpern H. The use of beams eye view volumetrics in the selection of non-coplanar radiation portals. Int J Radiat Oncol Biol Phys 1992;23:153–163. [47] Cho BCJ, Roa HW, Robinson D, Murray B. The development of target-eye-view maps for selection of coplanar or noncoplanar beams in conformal radiotherapy treatment planning. Medical Physics 1999;26:2367–2372. [48] Goitein M, Abrams M, Rowell D, Pollari H, Wiles J. Multi-dimensional treatment planning: Ii. beams eye-view, back projection, and projection through CT sections. Int J Radiat Oncol Biol Phys 1983;9:789–97. [49] Pugachev A, Xing L. Computer-assisted selection of coplanar beam orientations in intensity-modulated radiation therapy. Phys Med Biol 2001;46:2467–2476. [50] Pugachev A, Xing L. Pseudo beam’s-eye-view as applied to beam orientation selection in intensity-modulated radiation therapy. Int J Radiat Oncol Biol Phys 2001; 51:1361–1370. [51] Holder A, Salter B. A tutorial on radiation oncology and optimization. In H Greenberg, ed., Tutorials on Emerging Methodologies and Applications in Operations Research. Kluwer Academic Press, Boston, MA, 2004. [52] Morrill SM, Lane RG, Jacobson G, Rosen II. Treatment planning optimization using constrained simulated annealing. Phys Med Biol 1991;36:1341–61. [53] Oldham M, Khoo V, Rowbottom CG, Bedford J, Webb S. A case study comparing the relative benefit of optimising beam-weights, wedge-angles, beam orientations and tomotherapy in stereotactic radiotherapy of the brain. Phys Med Biol 1998; 43:2123–46. [54] Rowbottom CG, Webb S, Oldham M. Improvements in prostate radiotherapy from the customization of beam directions. Medical Physics 1998;25:1171–1179.

144

[55] Wang X, Zhang X, Dong L, Lie H, Wu Q, Mohan R. Development of methods for beam angle optimization for IMRT using an accelerated exhaustive search strategy. Int J Radiat Oncol Biol Phys 2004;60:1325–37. [56] Wang X, Zhang X, Dong L, Liu H, Gillin M, Ahamad A, Ang K, Mohan R. Effectiveness of noncoplanar IMRT planning using a parallelized multiresolution beam angle optimization method for paranasal sinus carcinoma. Int J Radiat Oncol Biol Phys 2005;63:594–601. [57] Woudstra E, Heijman BJM. Automated beam angle and weight selection in radiotherapy treatment planning applied to pancreas tumors. Int J Radiat Oncol Biol Phys 2004;56:878–88. [58] D’Souza WD, Meyer RR, Shi L. Selection of beam orientations in intensity-modulated radiation therapy using single-beam indices and integer programming. Phys Med Biol 2004;49:3465–3481. [59] Ehrgott M, Johnston R. Optimisation of beam directions in intensity modulated radiation therapy planning. OR Spectrum 2003;25:251–264. [60] Lim J, Ferris M, Shepard D, Wright S, Earl M. An optimization framework for conformal radiation treatment planning. INFORMS Journal On Computing 2006. [61] Wang C, Dai J, Hu Y. Optimization of beam orientations and beam weights for conformal radiotherapy using mixed integer programming. Phys Med Biol 2003; 48:4065–4076. [62] Fox C, Romeijn HE, Dempsey JF. Fast voxel and polygon ray-tracing algorithms for IMRT treatment planning, 2005. Submitted to Medical Physics. [63] Siddon RL. Prism representation: a 3d ray-tracing algorithm for radiotherapy applications. Phys Med Biol 1985;8:817–824. [64] Siddon RL. Fast calculation of the exact radiological path for a three-dimensional CT array. Medical Physics 1985;12:252–255. [65] Jacobs F, Sundermann E, Sutter BD, Christiaens M, Lemahieu I. A fast algorithm to calculate the exact radiological path through a pixel or voxel space. Journal of Computing and Information Technology (CIT) 1998;6:89–94. [66] Aleman DM, Romeijn HE, Dempsey JF. Beam orientation optimization methods in intensity modulated radiation therapy treatment planning. IIE Conference Proceedings 2006. [67] Aleman DM, Romeijn HE, Dempsey JF. A response surface approach to beam orientation optimization in intensity modulated radiation therapy treatment planning. In review 2006.

145

[68] Jones DR. A taxonomy of global optimization methods based on response surfaces. Journal of Global Optimization 2001;21:345–383. [69] Jones DR, Schonlau M, Welch WJ. Efficient global optimization of expensive black-box functions. Journal of Global Optimization 1998;13:455–492. [70] Csallner AE, Csendes T, Markót MC. Multisection in interval branch-and-bound methods for global optimization i. theoretical results. Journal of Global Optimization 2000;16:371–392. [71] Lagouanelle J, Soubry G. Optimal multisections in interval branch-and-bound methods of global optimization. Journal of Global Optimization 2004;30:23–38. [72] Epperly TGW, Pistikopoulos EN. A reduced space branch and bound algorithm for global optimization. Journal of Global Optimization 1997;11:287–311. [73] Barrientos O, Correa R. An algorithm for global minimization of linearly constrained quadratic functions. Journal of Global Optimization 2000;16:77–93. [74] Thoai NV. Convergence of duality bound method in partly convex programming. Journal of Global Optimization 2002;22:263–270. [75] Tuy H. On solving nonconvex optimization problems by reducing the duality gap. Journal of Global Optimization 2005;32:349–365. [76] Phong TQ, An LTH, Tao PD. Decomposition branch and bound method for globally solving linearly constrained indefinite quadratic minimization problems. Operations Research Letters 1995;17:215–220. [77] Bomze I. Branch-and-bound approaches to standard quadratic optimization problems. Journal of Global Optimization 2002;2:17–37. [78] Cambini R, Sodini C. Decomposition methods for solving nonconvex quadratic programs via branch and bound. Journal of Global Optimization 2005;33:313–336. [79] Aleman DM, Kumar A, Ahuja RK, Romeijn HE, Dempsey JF. Neighborhood search approaches to beam orientation optimization in intensity modulated radiation therapy treatment planning. in review 2007. [80] Kumar A. Novel methods for intensity-modulated radiation therapy treatment planning. Ph.D. thesis, University of Florida, 2005. [81] Geman S, Geman D. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 1984;6:721–741.

146

[82] Gelfand AE, Smith AFM. Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association 1990;85:398–409. [83] Smith RL. A monte carlo procedure for the random generation of feasible solutions to mathematical programming problems. Bulletin of the TIMS/ORSA Joint National Meeting 1980;:101. [84] Bélisle CJP, Romeijn HE, Smith RL. Hit-and-run algorithms for generating multivariate distributions. Mathematics of Operations Research 1993;18:255–266. [85] Kirkpatrick S, Gelatt CD. Optimization by simulated annealing. Science 1983; 220:671–680. [86] Bomze I. Fast simulated annealing. Physics Letters 1987;122A:157–162. [87] Bélisle CJP. Convergence theorems for a class of simulated annealing algorithms on Rd . Journal of Applied Probability 1992;29:885–895. [88] Aleman DM, Glaser D, Romeijn HE, Dempsey JF. A primal-dual interior point algorithm for fluence map optimization in intensity modulated radiation therapy treatment planning. work in progress 2007. [89] Jele´ n U, Alber M. A finite size pencil beam algorithm for IMRT dose optimization: density corrections. Physics in Medicine and Biology 2007;52:617–633. [90] Jele´ n U, Söhn M, Alber M. A finite size pencil beam for IMRT dose optimization. Physics in Medicine and Biology 2005;50:1747–1766. [91] Sempau J, Wilderman SJ, Bielajew AF. Dpm, a fast, accurate monte carlo code optimized for photon and electron radiotherapy treatment planning dose calculations. Phys Med Biol 2000;45:2263–91. [92] Baró J, Sempau J, Fernández-Varea JM, Salvat F. Penelope: An algorithm for monte carlo simulation of the penetration and energy loss of electrons and positrons in matter. Nuclear Instruments and Methods 1995;B100:31–46. [93] Sempau J, Baró J, Fernández-Varea JM, Salvat F. An algorithm for monte carlo simulation of coupled electron-photon showers. Nuclear Instruments and Methods 1997;B132:377–90. [94] Khan FM. The Physics of Radiation Therapy. Lippincott William and Wilkins, 1994.

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BIOGRAPHICAL SKETCH Dionne M. Aleman completed her bachelor’s degree in industrial and systems engineering at the University of Florida. She went on to study intensity modulated radiation therapy (IMRT) treatment planning optimization in the graduate program of the Department of Industrial and Systems Engineering at the University of Florida. She will receive her Doctor of Philosophy in Industrial and Systems Engineering in December of 2007, after which she will pursue a career in the Department of Mechanical and Industrial Engineering at the University of Toronto. Dionne plans to continue her research in cancer treatments, as well as other applications of operations research techniques to the medical and healthcare industries.

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