OPTIMIZATION MODELS FOR RADIATION THERAPY: TREATMENT PLANNING AND PATIENT SCHEDULING

Transcription

1 OPTIMIZATION MODELS FOR RADIATION THERAPY: TREATMENT PLANNING AND PATIENT SCHEDULING By CHUNHUA MEN 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

2 c 2009 Chunhua Men 2

3 To my wonderful family 3

4 ACKNOWLEDGMENTS I would like to thank all of those people who helped make this dissertation possible. First, I wish to thank my advisor, Dr. H. Edwin Romeijn for all his guidance, encouragement, support, and patience. Also, I would like to thank my committee members Dr. James F. Dempsey, Dr. Joseph P. Geunes, Dr. Stanislav Uryasev, and Dr. Fazil T. Najafi for their very helpful insights, comments and suggestions. Additionally, I would like to acknowledge Z. Caner Taşkın and Ehsan Salari who provided technical support and assistance with my projects. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION AN EXACT APPROACH TO DIRECT APERTURE OPTIMIZATION IN IMRT TREATMENT PLANNING Introduction Direct Aperture Optimization Column Generation Algorithm Introduction Derivation of the Pricing Problem Solving the Pricing Problem Results Clinical Problem Instances Dose-Volume Histogram (DVH) Criteria Stopping Rules Results Delivery efficiency Transmission effects Concluding Remarks NEW MODELS FOR INCORPORATING INTERFRACTION MOTION IN FLUENCE MAP OPTIMIZATION Introduction Beamlet-based Stochastic Optimization Models Beamlet-based Deterministic Model First New Stochastic Optimization Model Second New Stochastic Optimization Model Results Clinical problem instances Results Aperture-based Stochastic Optimization Models

6 3.3.1 Aperture-based Stochastic Optimization Models Derivation of the Pricing Problem Results Concluding Remarks OPTIMIZATION MODELS FOR PATIENT SCHEDULING PROBLEMS IN PROTON THERAPY DELIVERY Introduction Building the Strategic Models The Objective Function The Constraints The capacity The patient mix New patients treatment starting-time Anesthesia cases and twice-a-day fractions Gantry specialization and gantry switching The Optimization Model The Model with Penalty Function Results Input data Results for the basic scenario Sensitivity analysis Concluding Remarks A Heuristic Approach to On-line Patient (Re-)scheduling Problem Definition Model Description Sequence and timing decomposition Union sequence Restricted and desired time windows Evaluation criteria for the union sequence Solution Approach Timing optimizer for a single day Snout change index (SCI) Patient dissatisfaction index (P DI) Sequence optimizer Fitting a new patient Numerical example Concluding Remarks REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 2-1 Model dimensions Number of apertures without transmission effects Beam-on time without transmission effects Aperture-based FMO: number of apertures with transmission effects Aperture-based FMO: beam-on time with transmission effects Aperture-based FMO: DVH criteria under C1 without transmission effects Aperture-based FMO: DVH criteria under C1 with transmission effects added after optimization Aperture-based FMO: DVH criteria under C1 with transmission effects Beamlet-based FMO: DVH criteria under C1 without transmission effects Beamlet-based FMO: DVH criteria under C1 with transmission effects Model dimensions Radiation Therapy Oncology Group (RTOG P0126) criteria for prostate cancer Hotspots (%) for three models DVH criteria for critical structures (%) CPU running time (seconds) for three models (%) C1: Number of apertures and beam-on time C2: Number of apertures and beam-on time C3: Number of apertures and beam-on time C4: Number of apertures and beam-on time Patients Classification The results for the basic scenario model Gantry utilization per day using IP model (%) The definitions of scenarios

8 4-5 Sensitivity analysis Patient mix scenarios (%) Scenarios 17 19: Gantry available time (minutes) Scenario 17 19: Gantry utilization (%) Scenario 20: Revenue per fraction Results for Scenario Scenario 22: Patient mix for different anesthesia availabilities Scenario 22: Other statistics for different anesthesia availabilities Scenario 23: Patient mix for different anesthesia availabilities Scenario 23: Other statistics for different anesthesia availabilities Scenario 24: Patient mix for different anesthesia availabilities Scenario 24: Other statistics for different anesthesia availabilities Scenario 25: Patient mix for different anesthesia availabilities Scenario 25: Other statistics for different anesthesia availabilities Scenario 26: Patient mix for different anesthesia availabilities Scenario 26: Other statistics for different anesthesia availabilities Results for scheduling a new patient on different gantries

9 Figure LIST OF FIGURES page 1-1 (a) Targets and critical structures delineated on a slice of a CT scan; (b) Radiation beams passing through a patient (a) A multileaf collimator system 1 ; (b) the projection of an aperture onto a patient A typical CT slice illustrating target and critical structure deliniation. In particular, the targets PTV1 and PTV2 are shown, as well as the right parotid gland (RPG), the spinal cord (SC), and normal tissue (Tissue) DVHs of the optimal treatment plan obtained for Case 5 with C1 aperture constraints and the Convergence stopping rule Isodose curves for 73.8 Gy, 54 Gy, and 30 Gy on a typical CT slice, corresponding to the optimal treatment plan obtained for Case 5 with C1 aperture constraints and the Convergence stopping rule Case 5: The relative volume of (a) PTV1 and (b) PTV2 that receives in excess of its prescription dose The relative volume of (a) LSG and (b) RSG that receives in excess of 30 Gy The probability that a specified percentage of the target received the prescription dose for Case 1 (first stochastic model) The probability that a specified percentage of the target received the prescription dose for Case 1 (second stochastic model) The probability that a specified percentage of the target received the prescription dose for Case The probability that a specified percentage of the target received the prescription dose for Case The probability that a specified percentage of the target received the prescription dose for Case The probability that a specified percentage of the target received the prescription dose for Case The probability that a specified percentage of the target received the prescription dose for Case DVHs for critical structures (first stochastic model vs. traditional model)

10 3-9 DVHs for critical structures (second stochastic model vs. traditional model) DVHs for Case 1 in Scenario 135 (first model vs. traditional model) DVHs for Case 1 in Scenario 135 (second model vs. traditional model) Perceived and actual DVHs for critical structures for Case 1 and Case Perceived and actual DVHs for target Target dose coverage for two models with two stopping rules

11 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 MODELS FOR RADIATION THERAPY: TREATMENT PLANNING AND PATIENT SCHEDULING Chair: H. Edwin Romeijn Major: Industrial and Systems Engineering By Chunhua Men December 2009 My research addressed optimization models for radiation therapy treatment planning and patient scheduling. In intensity modulated radiation therapy (IMRT) treatment planning problems, I use direct aperture optimization (DAO) that explicitly formulates the fluence map optimization (FMO) problem as a convex optimization problem in terms of all multileaf collimator (MLC) deliverable apertures and their associated intensities and solve it using column generation method. In addition, the interfraction motion has been incorporated to the stochastic-programming based FMO and DAO models. Optimization models for patient scheduling problems in proton therapy delivery have also been studied in this research. 11

12 CHAPTER 1 INTRODUCTION Every year, approximately 300,000 people in the U.S. that are newly diagnosed with cancer may benefit from radiation therapy (American Cancer Society [2]). With this treatment modality, external beams of radiation pass through a patient with the aim of killing cancerous cells and thereby curing the patient. However, radiation kills both cancerous and normal cells. Many patients who are initially considered curable die of their disease and others may suffer unintended side effects from radiation therapy. The major reason is that radiation therapy treatment plans often deliver too little radiation dose to the cancerous cells, too much radiation dose to healthy organs, or both. The goal of radiation therapy treatment planning is therefore to design a treatment plan that delivers a prescribed dose to regions in the patient that contain (or are suspected to contain) cancerous cells (often called targets), while sparing nearby functional organs (often called critical structures). Figure 1-1(a) shows a slice of a CT scan on which several targets (PTV1 and PTV2) and critical structures (spinal cord (SC), right parotid gland (RPG)), and tissue are delineated. Typically, there are several clinical targets that we wish to irradiate and several critical structures that we wish to spare. It may be possible to eradicate the disease with a single beam of radiation. However, such a treatment may significantly damage normal cells in critical structures located along the path of the beam. Hence, multiple beams (usually 3 9) are used, whose intersection provides a high dose, whereas regions covered by a single beam or only a few beams receive much lower radiation doses (see Figure 1-1(b)). In particular, patients receiving radiation therapy are typically treated on a clinical radiation-delivery device called a linear accelerator, which can rotate around the patient. Conformal radiation therapy seeks to conform the geometric shape of the delivered radiation dose as closely as possible to that of the intended targets. In conventional conformal radiation therapy, this usually means 12

13 (a) (b) Figure 1-1. (a) Targets and critical structures delineated on a slice of a CT scan; (b) Radiation beams passing through a patient. that from each beam direction we deliver a single beam with uniform intensity level whose shape conforms to the beam s eye view of the targets in the patient as seen from that beam. Since the geometry of the targets and critical structure is different in each patient, customized apertures have to be manufactured for each patient. A relatively recent radiation therapy treatment delivery technique is Intensity Modulated Radiation Therapy (IMRT). With this technique, which has been employed in clinics since 1994, the linear accelerator is equipped with a so-called multileaf collimator (MLC) system. This system is able to block different parts of the beam, so that it can dynamically form a large number of different apertures (see Figure 1-2). IMRT thus allows for the delivery of a treatment plan that uses a much larger number of different apertures than conventional conformal radiation therapy, and thus the creation of very complex nonuniform dose distributions that deliver sufficiently high radiation doses to targets while limiting the radiation dose 1 Varian Medical Systems; 13

14 (a) (b) Figure 1-2. (a) A multileaf collimator system 1 ; (b) the projection of an aperture onto a patient. delivered to healthy tissues. The advent of IMRT has dramatically improved treatment efficiency and quality. Traditionally, IMRT treatment plans are developed using a two-stage process. In particular, we model each beam as a collection of hundreds of small beamlets (or bixels), and consider the intensities of each of these beamlets to be controllable on an individual basis. The problem of finding an optimal intensity profile (also called fluence map) for each beam is called the (beamlet-based) fluence map optimization (FMO) problem. The goal of this problem is to develop a treatment plan that satisfies and/or optimizes several clinical treatment plan evaluation criteria. However, this fluence map must then be followed by a leaf-sequencing stage in which the fluence maps are decomposed into a manageable number of apertures that are deliverable using a multileaf collimator (MLC) system. The objective of this second stage problem is to accurately reproduce the ideal fluence map while limiting the total treatment time. More formally, in this second stage it is desirable to limit both the total time that radiation is delivered, i.e., the total beam-on-time, and the total number of apertures used. In Chapter 2, we consider the problem of IMRT treatment planning using an approach which integrates 14

15 the beamlet-based FMO and leaf-sequencing problem into a single optimization model, which is usually referred to as a direct aperture optimization approach to FMO.We extend such a model to account for so-called transmission effects that have previously been dealt with in a post-processing phase, and we perform an extensive clinical evaluation of the approach and of different MLC delivery constraints. In general, the IMRT treatment plans are delivered during 5-7 weeks and a daily treatment is called a fraction. During a course of fractionated radiation therapy and between the fractions the organs of the patients body may move. Organ movements are divided into two general categories: interfraction motion and intrafraction motion ([36]). Interfraction organ motion consists of day-to-day changes, such as set-up errors, tumor shrinkage, weight lost, etc. Intrafraction motion refers to the internal organ motion occurring during the actual radiation treatment due to breathing, swallowing, etc. Traditionally, the IMRT treatment systems are based on a static patient model, which relies on a single static planning computed tomography (CT) image for treatment planning and evaluation. To account for organ motion, the conventional method is to add a margin around the clinical tumor volume (CTV) to get a planning target volume (PTV)([26, 27]). Instead of using a margin, in Chapter 3, we introduce new stochastic-programming based models for incorporating interfraction motion in FMO. Finally, a challenge of radiation therapy has been to efficiently deliver high quality treatment using limited and expensive resources. Effective scheduling systems have the conflicting goals of maximizing the utilization of resources and the number of patients, and minimizing the patient waiting time. The treatment providers are under pressure to reduce the cost and improve treatment quality. In recent years, patient scheduling is gradually becoming an essential component in medical care. However, little attention has focused on patient scheduling in radiation therapy. In Chapter 4, we develop optimization 15

16 models for strategic and operational patient scheduling problems in proton therapy delivery and test them using the data from the University of Florida Proton Therapy Institute (UFPTI). The overall goals of the strategic models are to analyze the treatment capacity and determine potential ways in which operations can be streamlined to increase this capacity. In addition, a new heuristic method has been developed and implemented to schedule patients for treatment delivery and the results indicate that we can obtain high-quality schedules in real-time. 16

17 CHAPTER 2 AN EXACT APPROACH TO DIRECT APERTURE OPTIMIZATION IN IMRT TREATMENT PLANNING 2.1 Introduction Traditionally, IMRT treatment plans are developed using a two-stage process: fluence map optimization is followed by a leaf-sequencing problem. Both the beamlet-based FMO problem and the leaf-sequencing problem are well-studied in the literature. For modeling and solution approaches to the FMO problem we refer to the review paper by Shepard et al. [64]. More recently, Lee et al. [40, 41] studied mixed-integer programming approaches, Romeijn et al. [61] proposed new convex programming models, and Hamacher and Küfer [24] and Küfer et al. [37] considered a multi-criteria approach to the problem. The problem of leaf-sequencing while minimizing total beam-on-time is very efficiently solvable in general. We refer in particular to Ahuja and Hamacher [1], Bortfeld [11], Kamath et al. [31], and Siochi [66]; in addition, Baatar et al. [5], Boland et al. [10], Kamath et al. [32, 33, 34, 35], Lenzen [42], and Siochi [66] study the problem under additional MLC hardware constraints, while Kalinowski [30] studies the benefits of allowing rotation of the MLC head. The problem of decomposing a fluence map into the minimum number of apertures has been shown to be strongly NP-hard (see Baatar et al. [5]), leading to the development of a large number of heuristics for solving this problem. Notable examples are the heuristics proposed by Baatar et al. [5] (who also identify some polynomially solvable special cases), Dai and Zhu [20], Que [54], Siochi [66], and Xia and Verhey [70]. In addition, Engel [23], Kalinowski [29], and Lim and Choi [43] developed heuristics to minimize the number of apertures while constraining the total beam-on-time to be minimal. Langer et al. [39] developed a mixed-integer programming formulation of the problem, while Kalinowski [28] proposed an exact dynamic programming approach. 17

18 Finally, Taşkın et al. [68] propose a new exact optimization approach to the problem of minimizing total treatment time. A major drawback of the decoupling of the treatment planning problem into a beamlet-based FMO problem and a MLC leaf sequencing problem is that there is a potential loss of treatment quality. This has led to the development of approaches that integrate the beamlet-based FMO and leaf-sequencing problems into a single optimization model, which are usually referred to as direct aperture optimization approaches to FMO. In this approach, we explicitly solve for a set of apertures and corresponding intensities in a single aperture-based FMO problem. For examples of integrated approaches to fluence map optimization, sometimes also called aperture modulation or aperture-based fluence map optimization, we refer to Preciado-Walters et al. [52], Shepard et al. [63], Siebers et al. [65], Bednarz et al. [8] and Romeijn et al. [60]. The way the dose distribution received by the patient is modeled in a beamlet-based FMO model is necessarily an approximation since this distribution depends not only on the intensity profile but also on the actual apertures used to deliver this profile. The current literature on aperture modulation has, however, has not yet exploited the ability of aperture modulation to take into account such effects. In particular, while the leaves in the MLC system do block most of the radiation beam, there is some small but not insignificant amount of dose (on the order of 1.5 2%, see Arnfield et al. [4]) that is transmitted through the leaves in the MLC system. Finally, while several aperture-based FMO approaches attempt to limit total treatment time by limiting the number of apertures used, these models do not explicitly incorporate total beam-on-time as a measure of treatment plan efficiency. In this chapter, we extend the approach developed by Romeijn et al. [60] by (i) allowing for the incorporation of more general treatment plan evaluation criteria and treatment plan constraints; (ii) accounting for transmission effects. In addition, we also 18

19 extend the method to MLC systems that can only deliver apertures that are rectangular in shape. Our goals are to evaluate the ability of our approach to efficiently find high-quality treatment plans with a limited number of apertures; evaluate the effect of MLC deliverability constraints on the required number of apertures and beam-on-time; evaluate the importance of explicitly incorporating transmission effects. 2.2 Direct Aperture Optimization With most forms of external-beam radiation therapy, a patient is irradiated from several different directions, which we assume are chosen based on experience by a physician or clinician. We will denote the set of beam directions by B. Each beam b B is discretized into a matrix of bixels, indicated by the set N b. For convenience we let N b B N b denote the set of all bixels. We will denote the set of apertures that can be delivered by a MLC system from beam direction b B by K b and the set of all deliverable apertures by K b B K b. For convenience, we let b k denote the beam that contains aperture k, i.e., k K bk for all k K. Clearly, each aperture can then be viewed as a subset of bixels in a beam, so we will denote a particular aperture k K b by the set of beamlets A k N b. With each aperture k K we associate a decision variable y k that indicates the intensity of that aperture. The dose distribution in a patient is evaluated on a discretization of the 3-dimensional geometry of the patient, obtained via a CT scan, into a number of voxels. We denote the set of all voxels by V, and associate a decision variable z j with each voxel j V that indicates the dose received by that voxel. The vector of voxel doses can be expressed as a linear function of the intensities of the apertures through the so-called dose deposition coefficients D kj, the dose received by voxel j V from aperture k K at unit intensity. 19

20 Finally, we assume that a collection, say L, of treatment plan evaluation criteria has been identified that measure clinical treatment plan quality and are expressed as functions of the dose distribution: G l : R V R for l L. Each of these criteria is usually, but not necessarily, a function of the dose distribution in a particular structure only. Without loss of generality we assume that the criteria are expressed in such a way that smaller values are preferred to larger values. Finally, we assume that all criteria are convex. This is justified by the fact that most criteria proposed in the literature to date are indeed convex or can be replaced by a convex criterion that has the property that the Pareto-efficient frontier associated with all criteria is unchanged (see Romeijn et al. [62]). Examples of such criteria are tumor control probability (TCP), normal tissue complication probability (NTCP), equivalent uniform dose (EUD), conditional value-at-risk (CVaR), voxel-based penalty functions, etc. (see, e.g., Niemierko [49] and [50], Lu and Chin [45], Kutcher and Burman [38], Rockafellar and Uryasev [58]). Our aperture-based FMO model can now be formulated as follows: minimize l L G l (z) subject to (A) z j = k K D kj y k for all j V (2 1) y k 0 for all k K. (2 2) Here z R V and y R K are the vectors containing the voxel doses and aperture intensities, respectively. Many other aperture-based FMO models that have been proposed in the literature are heuristics that are based on deterministic or stochastic search, such as simulated annealing, for which it often cannot be guaranteed that all deliverable apertures 20

21 are (explicitly or implicitly) considered. In contrast, our approach explicitly incorporates all deliverable apertures and corresponding intensities. Traditional beamlet-based FMO models as well as all aperture-based FMO models to date have assumed that the dose deposition coefficients can be written as D kj = i A k D ij (2 3) where D ij is the dose received by voxel j from bixel i at unit intensity. However, this definition ignores any transmission and scatter effects that are due to the shape of the apertures used. Both of these effects cannot be modeled in a beamlet-based FMO model. We will explicitly incorporate the transmission effect. In particular, the expression for the dose deposition coefficients given in (2 3) assumes that any bixel that is blocked in an aperture does not transmit any radiation. If we denote the fraction of dose that is transmitted by ɛ [0, 1], we obtain the following expression for the dose deposition coefficients: D kj = D ij + ɛ D ij i A k i N bk \A k = (1 ɛ) D ij + ɛ i A k = (1 ɛ) i A k D ij + ɛ D bk j i N bk D ij where D bj = i N b D ij. Clearly, the traditional expression (2 3) corresponds to the special case where ɛ = Column Generation Algorithm Introduction It is clear that the number of allowable apertures (i.e., the cardinality of K) is typically enormous. For example, consider an MLC that allows all combinations of left and 21

22 right leaf settings. Even with a coarse bixel grid and 5 beams, this would yield more than deliverable apertures to consider. However, it is reasonable to expect that in the optimal solution to (A) only a relatively small number of apertures will actually have positive intensity. The challenge is therefore to identify a small but judiciously chosen set of apertures that yield a high-quality treatment plan. Since it does not seem possible to intuitively identify or characterize such a set of apertures for each individual patient, we use a formal column generation approach to solving the aperture-based FMO problem. This method starts by choosing a limited number of apertures, for example corresponding to a conformal plan, given by a set ˆK K. It then solves a restricted version of (A) using only that set of apertures. Next, an optimization subproblem, called the pricing problem, is solved. This (i) (ii) identifies one or more promising apertures that will improve the current solution when added to the collection of considered apertures; or concludes that no such apertures exist, and therefore the current solution is optimal. In case (i), we add the identified apertures to ˆK, re-optimize the new aperture-based FMO problem, and repeat the procedure. Intuitively, the pricing problem identifies those apertures for which the improvement of the objective function per unit intensity is largest (and therefore show promise for significantly improving the treatment plan). The very nature of our approach thus allows us to study the effect of adding apertures on the quality of the treatment plan, thereby enabling a sound trade-off between the number of apertures and treatment plan quality Derivation of the Pricing Problem Let us denote the dual multipliers associated with constraints (2 1) and (2 2) by π j (j V ) and ρ k (k K). The Karush-Kuhn-Tucker (KKT) optimality conditions (see, e.g., Bazaraa et al. [7]) for (A), which are necessary and sufficient for optimality because of the 22

23 convexity of the objective function and the linearity of the constraints, can be written as follows: π j = l L γ l G l (z) z j for all j V ρ k = j V D kj π j for all k K z j = k K D kj y k for all j V y k ρ k = 0 for all k K y k, ρ k 0 for all k K. Any solution to this system can be characterized by y 0 only: this vector then determines, in turn, z, π, and ρ. Let (ŷ; ˆπ, ˆρ) be an optimal pair of primal and dual solutions to a subproblem problem in which only apertures in the set ˆK K are considered, where we have set ŷ k = 0 for k K\ ˆK. We can then conclude that this solution is in fact optimal for (A) if and only if ˆρ k 0 for all k K (note that this inequality is satisfied for k ˆK by construction). In other words, if and only if the optimal solution to the following so-called pricing problem is nonnegative: minimize k K D kj ˆπ j. j V Since each aperture contains beamlets in a single beam only, we may alternatively solve a pricing problem for each individual beam b B: minimize k Kb D kj ˆπ j. j V 23

24 Now note that if k K b we have ((1 ɛ) i Ak D ij + ɛ D bj ) D kj ˆπ j = j V j V = (1 ɛ) D ij ˆπ j + ɛ D bj ˆπ j. j V i A k j V This then means that the current solution is optimal for (A) if and only if, for all b B, the optimal solution to the following optimization problem ( ) minimize k Kb D ij ˆπ j exceeds the threshold value i A k j V ɛ D bj ˆπ j. (2 4) 1 ɛ We can now justify the intuition behind the pricing problem and the column generation algorithm that was provided earlier: realizing that j V D ij ˆπ j measures the per-unit change in objective function value if the intensity of beamlet i is increased, it follows that the pricing problem for a given beam identifies the aperture with the property that the rate of improvement in objective function value, as the intensity of the aperture is increased, is largest among all deliverable apertures. Furthermore, this aperture is added to the model only if increasing the intensity of that aperture actually corresponds to an improvement in objective function value Solving the Pricing Problem j V We will consider the following four common sets of hardware constraints on the set of deliverable apertures: C1. Consecutiveness constraint This constraint simply corresponds to the fact that apertures are shaped by pairs of leaves, which means that, in each given bixel row, the exposed bixels should be consecutive. ˆπ j 24

25 C2. Interdigitation constraint This constraint says that, in addition to C1, the left leaf of a row cannot overlap with the right leaf of an adjacent row. C3. Connectedness constraint This constraint says that, in addition to C2, the bixel rows that contain at least one exposed bixel should be consecutive. C4. Rectangle constraint This constraint says that only rectangular apertures may be formed. Note that constraint C4 corresponds to the use of conventional jaws only. Recently, the viability of this delivery technique has been shown by Kim et al. [71] for treating prostate cancer and by Earl et al. [21] for treating pancreas, breast, and prostate cancer. Romeijn et al. [60] provide polynomial-time algorithms for solving the pricing problem corresponding to C1 C3. In particular, suppose that each beam is discretized into an m n matrix of bixels. They then show that the pricing problem for a particular beam can be solved in O(mn) time for C1 and in O(mn 4 ) time for C2 and C3. For completeness sake, we will briefly describe these algorithms below. Next, we will develop an efficient algorithm for solving the pricing problem under C4. It is easy to see that, under C1, the pricing problem decomposes by bixel row, i.e., we may find the optimal leaf settings for each row individually and then form the optimal aperture by simply combining these leaf settings. We are thus interested in finding, for each bixel row, a consecutive set of bixels for which the sum of their coefficients in the objective function of the pricing problem is minimal. We can find such a set of bixels by making a single pass, from left to right, through the n bixels in a given row and beam. In doing so, we should keeping track of (i) the sum of the coefficients for all bixels considered so far, and (ii) the maximum value of these sums encountered so far. Now note that, at any point in this procedure, the difference between these two is a candidate for the best 25

26 solution value found so far, so we simply identify the leaf setting that corresponds to the minimum value of this difference. (See also Bates and Constable [6] and Bentley [9].) The algorithm for identifying the optimal aperture to add under C2 and C3 is somewhat more complicated. For these two situations, we formulate the pricing problem as the shortest path problem in an appropriately defined network. In particular, we define a node corresponding to each potential leaf setting in each bixel row, i.e., (r; c 1, c 2 ) for r = 1,..., m and c 1, c 2 = 1,..., n with c 1 < c 2, where c 1 and c 2 denote the rightmost and leftmost blocked bixel in row r, respectively. In addition, we define so-called source and sink nodes representing the top and bottom of the aperture. We then create arcs between nodes that correspond to feasible combinations of leaf settings in adjacent bixel rows, and assign to any arc leading to a particular node a cost equal to the sum of all coefficients corresponding to exposed bixels. That is, under C2 we create arcs between a pair of nodes if the interdigitation constraint is satisfied. Under C3, we ignore all nodes that correspond to bixel rows in which all bixels are blocked; then, in addition to the arcs created for C2, we also create arcs from the source node to all other nodes and from all nodes to the sink node, representing the fact that leaf settings that block all bixels are allowed at the top and bottom of the aperture. We will next develop a polynomial-time algorithm for the pricing problem associated with C4. For convenience, let (b, r, c) denote the bixel in row r and column c of beam b (r = 1,..., m; c = 1,..., n; and b B). Moreover, let (b, r 1, r 2, c 1, c 2 ) represent a rectangular aperture in which r 1 and r 2 denote the first and the last row, while c 1 and c 2 denote the leftmost and rightmost column of bixels which form the rectangular aperture in beam b. We can then formulate the pricing problem for beam b under C4 as follows: ( ) r 2 c 2 minimize D (b,r,c)j ˆπ j r=r 1 c=c 1 j V 26

27 subject to r 1 < r 2 c 1 < c 2 r 1, r 2 {1,..., m} c 1, c 2 {1,..., n}. It is easy to see that we can employ the algorithm for C1 to construct an algorithm for C4. In particular, suppose that we fix the range of rows in the rectangular aperture as r 1 and r 2 (with, of course, r 1 < r 2 ). Then the problem reduces to ( c 2 r2 ) minimize D (b,r,c)j ˆπ j subject to c=c 1 r=r 1 j V c 1 < c 2 c 1, c 2 {1,..., n} which is precisely a pricing problem of the form as for C1 with respect to the sum of rows r 1,..., r 2 in the matrix of objective function coefficients. Therefore, given these aggregate objective coefficients we can solve the pricing problem for C4 in O(m 2 n) time by enumerating all O(m 2 ) possible collections of consecutive rows and selecting the best solution among these candidates. Now note that we can, for each value of r 1, determine all aggregate objective function coefficients in O(mn) time. This means that all objective function coefficients can be determined in O(m 2 n) time, and the running time of the entire algorithm is O(m 2 n). Clearly, if, for each beam b B, the best solution found has an objective function value that exceeds the corresponding threshold value (2 4) derived in Section 2.3.2, no 27

28 aperture can improve the current solution, which is therefore optimal for (A). Otherwise, adding the optimal aperture, say (b, r1, r2, c 1, c 2), to the set ˆK can improve the current solution. 2.4 Results Clinical Problem Instances To test our models, three-dimensional treatment planning data for ten head-and-neck cancer patients was exported from a commercial patient imaging and anatomy segmentation system at the University of Florida Department of Radiation Oncology. This data was then imported into the University of Florida Optimized Radiation Therapy (UFORT) treatment planning system and used to generate the data required by the models described above. For all ten cases, we designed plans using five equispaced 60 Co-beams. Note that this does not affect the optimization algorithm, which is, without modification, applicable to high-energy X-ray beams as well (for example, see Romeijn et al. [60] for results with 6MV photon beams). The five beams are evenly distributed around the patient with angles 0, 72, 144, 216, 288, respectively. The nominal size of each beam is cm 2. The beams are discretized into bixels of size 1 1 cm 2, yielding on the order of 1,600 bixels. However, we reduce the set of beamlets that actually need to be considered in the optimization by using the fact that the actual volume to be treated is usually significantly smaller. That is, for each beam, we identify a mask consisting of only those bixels that can help treat the targets, i.e., we identify the bixels for which the dose deposition coefficient D ij associated with at least one target voxel is nonzero. We then extend the mask to a rectangle of minimum size to ensure that all deliverable apertures from C1 C4 that can help treat the targets are considered. For all cases we generate a voxel grid with voxels of size mm 3. To decrease the size of problems, we use a voxel size 28

29 8 8 8 mm 3 for unspecified tissue in the optimization models. But when we evaluate the treatment quality, we take the size mm 3 for all voxels. Table 2-1 shows the problem dimensions for the ten cases. Table 2-1. Model dimensions. case # structures total # voxels # voxels in the models # bixels ,017 17, ,298 24,300 1, ,234 36,414 1, ,113 38,680 2, ,255 15,985 1, ,636 13, ,262 21,386 1, ,369 18,661 1, ,837 14, ,294 40,202 2,183 Each case contained two targets, which are referred to as Planning Target Volume 1 (PTV1) and Planning Target Volume 2 (PTV2). PTV1 consists the Gross Tumor Volume (GTV, which refers to the best clinical estimation of the exact areas of the primary tumor volume) expanded to account for both sub-clinical disease as well as daily setup errors and internal organ motion; PTV2 is a larger target that also contains high-risk nodal regions, again expanded to account for sub-clinical disease and setup errors and organ motion. PTV1 and PTV2 have prescription doses of 73.8 Gy and 54 Gy, respectively. Figure 2-1 shows an example of target deliniation. Our FMO model employed treatment plan evaluation criteria that are quadratic one-sided voxel-based penalties. In particular, denoting the set of targets by T the set of critical structures by C, the set of all structures by S = T C, and the set of voxels in 29

30 Figure 2-1. A typical CT slice illustrating target and critical structure deliniation. In particular, the targets PTV1 and PTV2 are shown, as well as the right parotid gland (RPG), the spinal cord (SC), and normal tissue (Tissue). structure s S by V s, we use the following treatment plan evaluation criteria: G s (z) = α s ( max{0, T V s s z j } ) 2 j V s G s+ (z) = β s ( max{0, zj T s + } ) 2 V s j V s s T (2 5) s S. (2 6) (Clearly, this means that the set of treatment plan evaluation criteria can be expressed as L = {s : s T } {s+ : s S}.) The coefficients α s and β s (s S) are nonnegative weights associated with the clinical treatment plan evaluation criteria. Criteria (2 5) penalize underdosing below the underdosing threshold T s in all targets s T, while criteria (2 6) penalize overdosing above the overdosing threshold T + s in all structures s S. We choose this model based on the fact that it, in our experience, can be solved very efficiently and yields high-quality treatment plans. However, recall that our algorithm can easily be applied to models that include other convex treatment plan evaluation criteria, such as voxel-based penalty functions with higher powers, or EUD. The resulting 30

31 model (A) was solved by our in-house primal-dual interior point algorithm. We tuned the problem parameters (underdose and overdose thresholds and criteria weights) by manual adjustment based on two of the ten patient cases and a beamlet-based FMO model. We then used this set of parameters to solve different variants of the beamlet-based and aperture-based FMO problem for all ten patient cases. Finally, all of our experiments were performed on a PC with a 3.4 GHz Intel Pentium IV processor and 2 GB of RAM, running under Windows XP. Our algorithms were implemented in Matlab Dose-Volume Histogram (DVH) Criteria A tool that is most commonly employed by physicians to judge the quality of a treatment plan is the so-called (cumulative) dose-volume histogram (DVH). This histogram specifies, for a given target or critical structure, the fraction of its volume that receives at least a certain amount of dose. Using the finite representation of all structures using voxels, the value of the DVH at dose d is precisely the fraction of voxels in a structure who receive a dose of at least d units. Our approach is to employ a convex, and therefore efficiently solvable, formulation of the FMO problem. However, we use clinical DVH criteria to objectively verify both the ability of our models to create clinically acceptable treatment plans as well as the robustness of the problem parameters (which, as mentioned above, are not tuned for individual patients). We use the DVH criteria used in the Department of Radiation Oncology at the University of Florida. These criteria are based on the current clinical guidelines formulated by the Radiation Therapy Oncology Group ([55], [56]): PTV1 At least 99% should receive 93% of the prescribed dose ( Gy) At least 95% should receive the prescribed dose (73.8 Gy) No more than 10% should be overdosed by more than 10% of prescribed dose ( Gy) 31

32 No more than 1% of PTV1 should be overdosed by more than 20% of prescribed dose ( Gy) PTV2 At least 99% should receive 93% of the prescribed dose ( Gy) At least 95% should receive the prescribed dose (54 Gy) Salivary glands (right and left parotid glands, right and left submandibular glands) No more than 50% of each gland should receive more than 30 Gy Other structures Tissue should receive less than 60 Gy Spinal cord should receive less than 45 Gy Mandible should receive less than 70 Gy Brain stem should receive less than 54 Gy Eye should receive less than 45 Gy Optic nerve should receive less than 50 Gy Optic chiasm should receive less than 55 Gy Stopping Rules We use the column generation algorithm to solve the aperture-based FMO problem. In our implementation, we identify in each iteration the best aperture (among all beams) that could improve the treatment plan quality, and add this aperture to the set of apertures currently under consideration. This approach allows us to make a trade-off between the number of apertures and the treatment plan quality. As the number of apertures increases, at some point the improvement in treatment plan quality may not be clinically significant anymore. Moreover, a high number of apertures in general leads to a high beam-on time. Hence, rather than allowing the column generation algorithm to formally converge, we propose to terminate the algorithm based on the observed 32

33 development of the clinical DVH criteria. In particular, we investigate the merits of the following two stopping rules: Convergence: This stopping rule is based on observing that treatment plan quality, with respect to a particular criterion, has not improved markedly in recent iterations. More formally, we say that we are satisfied with the solution with respect to a particular criterion if, in the last 5 iterations, the range of observed criterion values spans less than δ (where we use δ = 0.5% for target criteria and δ = 2% for critical structure criteria). Clinical: This stopping rule is based on observing that the treatment plan performance with respect to a clinical criterion has been satisfactory in the last 5 iterations. More formally, we say that we are satisfied with the solution with respect to a particular criterion if either the first stopping rule is satisfied or, in the last 5 iterations, the clinical DVH criterion is satisfied, allowing for a 1% error bar in all but one of those iterations. (Note that we allow for the clinical stopping rule to be satisfied if the convergence stopping rule is satisfied to account for the fact that certain clinical criteria cannot be achieved. This may, for example, be the case if a critical structure is wholly or partially contained in a target.) In either case, we say that the algorithm has converged if the stopping rule has been satisfied for all convergence (resp. clinical) criteria. Moreover, we report the solution obtained in the first iteration of the last sequence of 5 iterations Results We divide the discussion of our results into two parts according to the goals that we formulated in Section 1. First, we evaluate the ability of our approach to efficiently find high-quality treatment plans with a limited number of apertures, as well as the effect of MLC deliverability constraints on the required number of apertures and beam-on-time. Next, we evaluate the importance of explicitly incorporating transmission effects into the optimization model. However, before we do so, in Figures 2-2 and 2-3 we illustrate the behavior of our FMO model by showing the DVHs and isodose curves superimposed on a typical CT slice, both corresponding to an optimal treatment plan found for Case 5. The 33

34 isodose curves show both the conformality of the plan with respect to the two targets and the sparing of the spinal cord and saliva glands. Figure 2-2. DVHs of the optimal treatment plan obtained for Case 5 with C1 aperture constraints and the Convergence stopping rule. Figure 2-3. Isodose curves for 73.8 Gy, 54 Gy, and 30 Gy on a typical CT slice, corresponding to the optimal treatment plan obtained for Case 5 with C1 aperture constraints and the Convergence stopping rule. 34

35 Delivery efficiency Tables 2-2 and 2-3 show the number of apertures and beam-on-time (in minutes) for the ten cases obtained with our aperture-based approach. We used the two stopping rules described in Section For these experiments, we have not incorporated any transmission effects. For comparison purposes, the tables also show the results of traditional beamlet-based FMO where the optimal fluence maps were first discretized to integer multiples of 5% of the maximum beamlet intensity in each beam, and subsequently decomposed into apertures with the objective function of minimizing the beam-on time. We used the algorithms by Kamath et al. [31, 32] to minimize the beam-on-time under C1 and C2, respectively. Furthermore, we applied a modification of the approach by Boland et al. [10] to minimize the beam-on-time under C3, while we used a linear programming model for the case of C4. (Note that there are, in general, many decompositions that attain the minimum beam-on-time; the number of apertures that is given is for a particular solution that the so-called leaf-sequencing algorithm found, but that number is not minimized explicitly.) Our main conclusions are: With our direct aperture optimization approach and based on our stopping rules, we conclude that the number of apertures required under aperture constraints C1 C3 is, on average, on the order of 30 45, increasing to with jaws-only delivery. The direct aperture optimization approach leads to a reduction of the required number of apertures by, on average, more than 75% of the number obtained with the traditional two-phase approach. Moreover, the required beam-on-time is reduced by, on average, more than 50%. There is very little effect on the required number of apertures and beam-on-time when interdigitation or connectedness constraints are imposed; however, the required number of apertures and beam-on-time increases by, on average, approximately 50% when only rectangular apertures are considered. 35

36 It is interesting to see that in some cases fewer apertures are required when the deliverability constraints are strengthened. This is caused by a combination of the fact that our approach does not explicitly minimize the number of apertures and the fact that the effect of some constraints (in particular, the added constraint in C3 as compared to C2) is apparently negligible in practice. The amount of time required, on average, by our optimization algorithm to find the aperture based solutions ranges from about 1 3 minutes of CPU time for the case of consecutiveness constraints only. The required time increases to up to 4 minutes when interdigitation and connectedness are imposed, while it further increases to up to about 12 minutes when only rectangular apertures are allowed. This is in comparison with an average of a little over 1 minute of CPU time required for the traditional two-phase approach. Note that these times do not include the time required to read the DICOM data and compute the dose deposition constraints, which took about minutes of CPU time depending on the size of the case. However, note that these tasks only need to be performed once for each patient. To illustrate how our approach may be used to make a trade-off between treatment plan quality and delivery efficiency, Figures show the behavior of target coverage and submandibular gland sparing as a function of the number of apertures used for C1 C4 for a typical example, Case Transmission effects We have also studied the effect of incorporating transmission effects into the FMO problem. First, Tables 2-4 and 2-5 show the number of apertures and beam-on-time with incorporation of transmission effects. We have used ɛ = 1.7% as the transmission rate (see Arnfield et al. [4]). A comparison with Tables 2-2 and 2-3 reveals that the incorporation of transmission effects has very little effect on the treatment delivery efficiency. 36

37 Next, Tables show the results of the aperture-based FMO problem under C1 without and with transmission effects using the convergence stopping rule. In particular, the tables show values of the DVH criteria for the targets and main critical structures for all ten cases. For example, the data in the column labeled represents the fraction of volume (in %) of PTV2 receiving Gy. The labels of the other columns follow a similar format, where the acronyms correspond to the left parotid gland (LPG), right parotid gland (RPG), left submandibular gland (LSG), and right submandibular gland (RSG). Comparing Tables 2-6 and 2-8 suggests that treatment plans of very similar quality can be found with and without incorporation of transmission effects. However, the results of Table 2-6 neither incorporate transmission in the optimization problem nor account for transmission in the presentation of the actual results, so that the treatment plan quality in that table is a perceived rather than an actual one. Table 2-7 shows the actual quality of the treatment plan that was obtained when solving an optimization model that does not take transmission effects into account, i.e., for the results in Table 2-7 transmission effects were added a posteriori to the plan. Finally, we analyzed the results of using a beamlet-based FMO approach followed by a leaf sequencing phase, under C1. Table 2-9 shows the perceived quality of the obtained treatment plan (in which transmission effects are ignored), while Table 2-10 shows the actual quality of the treatment plan (in which transmission effects are added to the final treatment plan). From the latter two tables, it is clear that using a beamlet-based FMO optimization approach may severely underestimate target hotspots (overdosing) and effects on critical structures. The direct aperture optimization approach with transmission effects incorporated, however, provides a high-quality treatment plan with, on average, a comparable number of apertures and beam-on-time. Taking Case 5 as an example, a 37

38 treatment plan that ignored transmission effects appeared to spare all four saliva glands, while less than 10% of PTV1 received in excess of 110% of its the prescription dose. Adding transmission effects to the plan found using beamlet-based FMO showed that in fact only two saliva glands were spared with the PTV1 hotspot increasing to over 80%. Incorporating transmission effects into the aperture-based FMO model showed that it was possible to spare all saliva glands and that keep the dose to PTV1 in excess of 110% of its prescribed dose to about 3%. 2.5 Concluding Remarks In this chapter, we consider the problem of IMRT treatment planning using direct aperture optimization. We use an exact approach that explicitly formulates the FMO problem as a convex optimization problem in terms of all MLC deliverable apertures and their associated intensities. However, the number of deliverable apertures, and therefore the number of decision variables and constraints in the new problem formulation, is typically enormous. To overcome this, we use an iterative approach that employs a subproblem whose optimal solution either provides a suitable aperture to add to a given pool of allowable apertures or concludes that the current solution is optimal. We are able to handle standard consecutiveness, interdigitation, and connectedness constraints that may be imposed by the particular MLC system used, as well as jaws-only delivery. Our approach has the additional advantage that it can explicitly account for transmission of dose through the part of an aperture that is blocked by the MLC system, yielding a more precise assessment of the treatment plan than what is possible using a traditional beamlet-based FMO problem. Finally, we develop and test two stopping rules that can be used to identify treatment plans of high clinical quality that are deliverable very efficiently. Tests on clinical head-and-neck cancer cases showed the efficacy of our approach, yielding treatment plans comparable in quality to plans obtained by the traditional method with 38

39 a reduction of more than 75% in the number of apertures and a reduction of more than 50% in beam-on-time, with only a modest increase in computational effort. The results also show that delivery efficiency is very insensitive to the addition of traditional MLC constraints; however, jaws-only treatment requires about a doubling in beam-on-time and number of apertures used. Finally, we showed the importance of accounting for transmission effects when assessing or, preferably, optimizing treatment plan quality. 39

40 Table 2-2. Number of apertures without transmission effects. Aperture-based Beamlet-based case Clinical Convergence C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C Average Table 2-3. Beam-on time without transmission effects. Aperture-based Beamlet-based case Clinical Convergence C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C Average

41 Table 2-4. Aperture-based FMO: number of apertures with transmission effects. Clinical Convergence case C1 C2 C3 C4 C1 C2 C3 C Average Table 2-5. Aperture-based FMO: beam-on time with transmission effects. Clinical Convergence case C1 C2 C3 C4 C1 C2 C3 C Average Table 2-6. Aperture-based FMO: DVH criteria under C1 without transmission effects. PTV2 PTV2 PTV1 PTV1 PTV1 PTV1 LPG RPG LSG n/a n/a n/a n/a

42 Table 2-7. Aperture-based FMO: DVH criteria under C1 with transmission effects added after optimization. PTV2 PTV2 PTV1 PTV1 PTV1 PTV1 LPG RPG LSG n/a n/a n/a n/a Table 2-8. Aperture-based FMO: DVH criteria under C1 with transmission effects. PTV2 PTV2 PTV1 PTV1 PTV1 PTV1 LPG RPG LSG n/a n/a n/a n/a Table 2-9. Beamlet-based FMO: DVH criteria under C1 without transmission effects. PTV2 PTV2 PTV1 PTV1 PTV1 PTV1 LPG RPG LSG n/a n/a n/a n/a

43 (a) (b) Figure 2-4. Case 5: The relative volume of (a) PTV1 and (b) PTV2 that receives in excess of its prescription dose. Table Beamlet-based FMO: DVH criteria under C1 with transmission effects. PTV2 PTV2 PTV1 PTV1 PTV1 PTV1 LPG RPG LSG n/a n/a n/a n/a

44 (a) (b) Figure 2-5. The relative volume of (a) LSG and (b) RSG that receives in excess of 30 Gy. 44

45 CHAPTER 3 NEW MODELS FOR INCORPORATING INTERFRACTION MOTION IN FLUENCE MAP OPTIMIZATION 3.1 Introduction Geometrical uncertainties (interfraction and intrafraction motion) in radiotherapy treatments are likely to cause significant differences between the intended and actually delivered dose distribution in the patient. To account for these uncertainties, the conventional method is to add a margin to the so-called clinical target volume (CTV) to form the so-called planning target volume (PTV). CTV refers to the specific zone surrounding the gross tumor volume at high risk for clinical extension of the tumor and PTV refers to the specific zone surrounding the CTV with a margin accounting for the interfraction and intrafraction motion. Many studies (Antolak and Rosen[3], Stroom et al. [67], Herk et al. [25] and Parker et al. [51]) derived expressions for the size of margin. However, these margin-based methods have some theoretical and practical problems: the prescribed dose is applied to the entire PTV (rather than the CTV only), and the margin may not adequately model the changes in dose distributions due to the random deviations of organ motion: the CTV may only have a very small chance to reach some of the edges of this high dose area but in other directions it may extend beyond this area. Therefore, the CTV may not receive adequate dose while organs close to the CTV (critical structures or organs at-risk) may not be spared due to overdosing; moreover, the dose distribution of PTV cannot be used to accurately evaluate the treatment plan quality; finally, the planning dose is delivered over a number of fractions so the dose received by PTV is an accumulated dose. Hence, the actual dose received in each fraction may significantly differ from the planned one and the total dose may be badly estimated as well. Because of these problems of margin-based methods, attention has recently shifted to the development of alternative methods. The most common methods are to process the 45

46 static planned dose distribution by convolving it with a probability density function (PDF) which describes the motion uncertainties (Lujan et al. [46] and Carter and Beckham [47]). This dose-convolution method assumes shift invariance of the dose distribution. However, internal inhomogeneities and surface curvature may lead to violations of this assumption (Craig et al. [18]); In addition, this method assumes that the patient is treated with an infinite number of fractions, each delivering an infinitesimally small dose. The effect of finite fractionation appears to have a greater impact on the dose distribution than plan evaluation parameters (Craig et al. [19]). Recently, Lu et al. [44] showed that the motion effects can be accounted for by modifying the fluence maps. After such modification, dose calculation is the same as those based on a static planning image. However, this method is only suitable for the cases when the patient motion is small (Lu et al. [44]). Chan et al. [15] introduced a robust methodology for dealing with IMRT optimization problems under intrafractional uncertainty induced by breathing motion. They used the idea of a motion probability mass function (PMF) along with an associated set describing the uncertainty of this PMF as their model of data uncertainty. In Chapter 2, we introduced an aperture-based deterministic FMO model which is actually a margin-based model. In this chapter, we will first focus on beamlet-based stochastic models and then move to aperture-based stochastic models and both stochastic models account for interfraction motion. We will not consider intrafraction motion which need to be analyzed separately. Our goals in this chapter are to build new beamlet-based stochastic optimization models accounting for interfraction motion (especially due to set-up errors); evaluate the ability of our models to find high-quality treatment plans; build and solve aperture-based stochastic models by direct aperture optimization approach. 46

47 3.2 Beamlet-based Stochastic Optimization Models Beamlet-based Deterministic Model Our beamlet-based stochastic optimization models were derived from the traditional beamlet-based deterministic FMO model, we therefore first introduce the traditional one. The related notation can be found in Chapter 2. Beamlet-based FMO is to find the optimal intensity (fluence map) for each beamlet and we will denote the intensity of beamlet i N by x i, then the vector of voxel doses can be expressed as a linear function of the intensities of the beamlets through dose deposition coefficients D ij. Then the beamlet-based deterministic model can be written as minimize l L G l (z) subject to z j = i N D ij x i for j V x i 0 for i N. Our FMO models employ treatment plan evaluation criteria that are quadratic one-sided voxel-based penalties (see equations 2 5 and 2 6). If we denote the set of target voxels by V T and the set of critical structure voxels by V C (V T V C = V ), then we can rewrite the criteria as (2 5) and (2 6) as F j (z j ) = α s ( max{0, T V s s z j } ) 2 F j (z j ) = β s ( max{0, zj T s + } ) 2 V s s T, j V T s S, j V 47

48 where G s (z) = j V s F j (z j ) s T, j V T G s+ (z) = j V s F j (z j ) s S, j V. Then the objective function can be re-formulated to minimize j V F j (z j ) z j = i N D ij x i for j V x i 0 for i N First New Stochastic Optimization Model If we could account for the uncertainties of interfraction motion in the FMO model, much more effective treatment plans may be obtained and implicit patient-dependent margins can be created to optimally hedge against geometry changes. Hence, we would be able to much more effectively balance the conflicting goals of tumor dose coverage and critical structure sparing. Integrating the interfraction motion into the FMO model essentially transforms the deterministic optimization problem into a stochastic one since D ij, the dose receive by each voxel at unit intensity, is no longer known with certainty. Since the current optimization models are already challenging large-scale optimization problems, models that explicitly account for motion may quickly become intractable. To simplify the model, we could (since the number of fractions is relatively large) consider replacing the coefficients D ij in the original model by their expected values E(D ij ). Implicitly, this assumes that the relevant quantity is the total dose received by each voxel over the course of the treatment. This assumption may be appropriate for critical 48

49 structures but it is inappropriate for targets, where a more important measure is the amount of dose received in each individual fraction. Hence, our model is to apply the simplifying assumption to the critical structures, but penalize the dose to targets in each fraction via the incorporation of a large number of scenarios. Since the number of critical structure voxels in general is much more than that of target voxels, our assumption could implicitly decrease the size of the stochastic model. The following new notation has been introduced: Dij denotes the expected (or average) dose received by voxel j from beamlet i at unit intensity; S denotes the number of scenarios; Dij s denotes the dose received by voxel j from beamlet i at unit intensity in scenario s = 1,..., S and zj s denotes the dose received by voxel j V T in scenario s = 1,..., S. Since the target in the traditional model (i.e., PTV) contains interfraction motion margin but the target in our new model does not (instead of adding margin, our model accounts for interfraction motion by generating scenarios), the target in our model is a subset of the PTV. Our stochastic FMO model is then formulated as follows: minimize 1 S S F j (zj s ) + F j (z j ) j V T j V C s=1 subject to z s j = i N D s ijx i for j V T, s = 1,..., S z j = i N D ij x i for j V C x i 0 for i N Second New Stochastic Optimization Model The target cannot be severely overdosed because (i) the critical structures around the target may be overdosed too; (ii) the tumor cannot be accurately specified and the target may contain health tissues and/or critical structures and (iii) traditionally, the prescribed 49

50 dose is applied to the entire PTV which contain very high volume of normal tissues and/or critical structures because a margin is added to the CTV. However, the target in our stochastic model is a subset of PTV and it potentially decreases the volume of heath organs inside of the target and then we can slightly loosen the the treatment criteria for the overdosing to target, and therefore we can deliver adequate dose to the tumor. In the first stochastic model, we penalize dose for both overdosing and underdosing to targets in each fraction, then in the second model, we separate the objective function into two parts: one is for measuring the treatment plan quality for all voxels which receive the averaged dose over all scenarios exceeding the prescribed one (denoted by F + j ); another is for measuring the treatment plan quality for the target voxels which receive the dose in each scenario less than the prescribed one (denoted by F j ). The model is a potentially better representation of the biological considerations. The new model can be formulated as follows: subject to minimize 1 S S F j (zs j ) + F + j (z j) j V T s=1 j V z s j = i N D s ijx i for j V T, s = 1,..., S z j = i N D ij x i for j V x i 0 for i N. Note that this method implicitly decreases the size of the problem because instead of evaluating the treatment quality for the overdosed target voxels for each scenario, the new model evaluates the overdosed target voxels based on averaged doses. Since this model put penalty for the underdosing target voxels in each scenario, most likely, adequate doses can be delivered to the target. 50

51 3.2.4 Results Clinical problem instances To test our models, we use five clinical cases of prostate cancer. Please refer to Chapter 2 for detail information about the sizes of bixes and voxels, how to import data, etc. For all five cases, we design plans using nine equispaced 60 Co-beams. The nine beams are evenly distributed around the patient with angles 0, 40, 80, 120, 160, 200, 240, 280, 320, respectively. There are one target (prostate) with prescription dose 73.8 Gy and three critical structures: bladder, rectum and femoral head. Table 3-1 shows the problem dimensions for the five cases. Table 3-1. Model dimensions. Case total # voxels # voxels in the models # bixels 1 205,911 35,988 2, ,216 22,330 2, ,324 38,731 2, ,628 29,828 2, ,184 40,636 2,736 Since our model could account for the interfraction motion, the target in the optimization model is the CTV with the intrafraction motion margin. Empirically, the standard deviation of set-up errors in each direction has been found to be approximately 3 mm (El-Bassiouni et al. [22], Britton et al. [12], Price [53] and Wang et al. [69]) and we therefore generated 200 scenarios by perturbing the location of the patient according to a normal distribution with a mean of 0 and a standard deviation of 3 mm in each coordinate direction for each case. For target, we test the dose coverage for individual scenarios; for critical structures, we check the averaged dose distribution over all scenarios. Dose-volume criteria and constraints have been established to keep toxicity at acceptable levels. However, PTV is the CTV with interfraction and intrafraction motion margins and the target in our model is CTV with intrafraction motion margin so the traditional 51

52 DVH criteria for PTV are not suitable for the target in our models. We therefore evaluate the target dose coverage for each individual scenario: we estimate the probability that a fraction β of the target receives the prescription dose, for different values of β. For critical structures, specifically for prostate radiation therapy, we use the Radiation Therapy Oncology Group criteria ([57]) (in Table 3-2) which are commonly used by physicians and researchers (Moiseenko et al. [48], Rodrigues et al. [59] and Chen et al. [16]). Finally, we compare our solutions with the ones obtained from the traditional method. The width of the set-up error margin is set to be about 1.65 times the standard deviation (see Antolak and Rosen [3]). Hence, 5 mm of margin in each direction has been added to the target for the traditional method. Table 3-2. Radiation Therapy Oncology Group (RTOG P0126) criteria for prostate cancer. Rectum Bladder 15% to receive 75 Gy 15% to receive 80 Gy 25% to receive 70 Gy 25% to receive 75 Gy 35% to receive 65 Gy 35% to receive 70 Gy 50% to receive 60 Gy 50% to receive 65 Gy Our FMO models employ treatment plan evaluation criteria that are quadratic one-sided voxel-based penalties. We tuned the problem parameters (underdose and overdose thresholds and criteria weights) by manual adjustment. In general, compared to the traditional model, we increase the penalty weights to the target dose distributions while decrease the penalty weights to the critical structures dose distributions. The models were solved by our in-house primal-dual interior point algorithm. All experiments were performed on a PC with 2.66 GHz Intel Quad CPU and 4 GB of RAM, running under Windows Vista. Our algorithms were implemented in Matlab Results We first test the effects of varying the number of scenarios that is included in the optimization models. A large number of scenarios will lead to unacceptable CPU 52

53 running time while a small number of scenarios may be insufficient to capture the actual uncertainties and may therefore result in unacceptable treatment quality. By testing different number of scenarios included in the models, we could find a balance between the computation effort and the treatment quality. We include 25, 50, 75 and 100 scenarios in the two optimization models and test the treatment qualities for scenarios which are in and out of the optimization models. Taking Case 1 as an example, Figure 3-1 and 3-2 estimate the probability that a specified percentage of the target received the prescription dose for two models, respectively. We observe that the treatment qualities from the scenarios out of the optimization models are not as good as ones included in the optimization models if we only include 25, 50 or 75 scenarios in the models. For instance, if we include 25 scenarios in the first optimization model, for all these 25 scenarios, at least 95% of target volume could receive the prescription dose, however, there are only 21 out of 25 scenarios (i.e., 84%) out of the optimization model in which at least 95% of target volume could receive the prescription dose. If we include 100 scenarios in the models, the treatment qualities obtained from the scenarios out of the optimization models are very similar with the ones obtained from the scenarios in the optimization models (see Figure 3-1 and 3-2). By testing all five cases, we conclude that 100 is the (estimated) minimal number of scenarios which should include in both optimization models in order to obtain the stable treatment quality for all available scenarios. Figures 3-3 to 3-7 estimate the probability that a specified percentage of the target received the prescription dose for Scenarios and , respectively, for the five cases. No remarkable differences are observed between the scenarios in and out of the optimization models. We observe that, in general, our stochastic models obtain better target dose coverage than the traditional model does. Taking Case 1 as an example, the probability of 95% target coverage is 98% for the first stochastic model; the probability 53

54 (a) (b) (c) (d) Figure 3-1. The probability that a specified percentage of the target received the prescription dose for Case 1 (first stochastic model). is 99% for the second stochastic model; while by solving the traditional model, that probability is only 91%. In particular, the second stochastic model obtain slightly better target dose coverage than the first one. Figure 3-8 and 3-9 show the DVHs for the critical structures for the two stochastic models compared to the traditional method, respectively. Table 3-4 lists the corresponding DVH values based on the criteria. We conclude from Figure 3-8, 3-9 and Table 3-4 that our methods could obtain better target dose coverage with (almost) no additional dose to the critical structures. Besides estimating the probability that a specified percentage of the target receive the prescription dose for the two models, we would like to check another two important 54

55 (a) (b) (c) (d) Figure 3-2. The probability that a specified percentage of the target received the prescription dose for Case 1 (second stochastic model). criteria: target coldspots (underdosing) and hotspots (overdosing) which refer to the volume of target which receives less than 93% and more than 110% of prescription dose, respectively. Table 3-3 shows the averaged hotspots for all five cases. We observe that even though the second stochastic model obtain better target dose coverage than the traditional and the first stochastic model, in general, it does not have higher hotspots. In most of scenarios, 100% of target receive at least 93% of prescription dose for all five cases so there are almost no coldspots for all models. Our stochastic optimization models take individual dose distributions for target into account and robust treatment plans can be obtained while the traditional method does not 55

56 have this merit: in some extreme scenarios, the treatment quality may be very bad using the traditional method. We would like to take Case 1 as an example to check the target dose coverage for Scenario 135. Note that this scenario is not included in the optimization models. Figure 3-10 and 3-11 show the DVHs for first and second stochastic models compared to the traditional method, respectively. We can see that about 96% and 98% of target receive the prescription dose (73.8 Gy) by solving the first and second stochastic models, respectively, while for the traditional method, that volume is only 92%. The amount of time required, on average, by our optimization algorithms to find the solutions ranges from about 3 7 minutes of CPU time for two models while it takes less than 2 minutes to solve the traditional model. Moreover, the time required by the second stochastic model is less than the first model due to the smaller size of the model. Table 3-5 records the CPU time for these three methods. Finally, recall that the traditional method does not account for interfraction motion in either the optimization model or its reporting of the treatment plan quality. Hence, the perceived treatment plan quality may differ from the actual one. To assess this effect, we compared the perceived DVHs reported from the traditional model with the actual DVHs that take into account the uncertainty using the 200 scenarios from the interfraction motion model. The results indicate that the perceived DVHs for critical structures are very close to the actual ones (see examples in Figure 3-12). However, the perceived DVHs overestimate the dose distribution to the targets (see results in Figure 3-13). This further underscores the importance of taking interfraction motion uncertainty into account explicitly. 56

57 (a) (b) Figure 3-3. The probability that a specified percentage of the target received the prescription dose for Case 1. (a) (b) Figure 3-4. The probability that a specified percentage of the target received the prescription dose for Case 2. (a) (b) Figure 3-5. The probability that a specified percentage of the target received the prescription dose for Case 3. 57

58 (a) (b) Figure 3-6. The probability that a specified percentage of the target received the prescription dose for Case 4. (a) (b) Figure 3-7. The probability that a specified percentage of the target received the prescription dose for Case 5. Table 3-3. Hotspots (%) for three models. Case traditional model stochastic model 1 stochastic model

59 (a) (b) (c) (d) (e) Figure 3-8. DVHs for critical structures (first stochastic model vs. traditional model). 59

60 (a) (b) (c) (d) (e) Figure 3-9. DVHs for critical structures (second stochastic model vs. traditional model). 60

61 (a) Figure DVHs for Case 1 in Scenario 135 (first model vs. traditional model). (a) Figure DVHs for Case 1 in Scenario 135 (second model vs. traditional model). 61

62 Table 3-4. DVH criteria for critical structures (%). Rectum Bladder Gy ( 15%) ( 25%) ( 35%) ( 50%) ( 15%) ( 25%) ( 35%) ( 50%) 1 Trad Stoch Stoch Trad Stoch Stoch Trad Stoch Stoch Trad Stoch Stoch Trad Stoch Stoch Table 3-5. CPU running time (seconds) for three models (%). Case traditional model stochastic model 1 stochastic model (a) (b) Figure Perceived and actual DVHs for critical structures for Case 1 and Case 2 62

63 (a) (b) (c) (d) (e) Figure Perceived and actual DVHs for target. 63

64 3.3 Aperture-based Stochastic Optimization Models Aperture-based Stochastic Optimization Models In Chapter 2, we used a direct aperture optimization approach to design radiation therapy treatment plans for individual patients which integrates the beamlet-based FMO and leaf-sequencing problems into a single optimization model. We can apply this approach to our beamlet-based stochastic optimization models. The related notation can be found in Chapter 2. The first aperture-based stochastic FMO model can be formulated as follows: subject to minimize 1 S S F j (zj s ) + F j (z j ) j V T j V C s=1 z s j = k K D s kjy k for j V T, s = 1,..., S (3 1) z j = D kj y k k K for j V C (3 2) y k 0 for all k K (3 3) where D s kj = i A k D s ij for j V T, s = 1,..., S D kj = i Ak D ij for j V C. If we incorporate the transmission effect, we obtain the following expression for the dose deposition coefficients: D s kj = (1 ɛ) i A k D s ij + ɛ D kj = (1 ɛ) i Ak i N bk D s ij D ij + ɛ Dij for j V C. i N bk for j V T, s = 1,..., S 64

65 The second aperture-based stochastic FMO model can be formulated as follows: minimize 1 S S F j (zs j ) + F + j (z j) j V T s=1 j V subject to z s j = k K D s kjy k for j V T, s = 1,..., S (3 4) z j = D kj y k k K for j V (3 5) y k 0 for all k K. (3 6) Derivation of the Pricing Problem In Chapter 2, we derived the pricing problem for the aperture-based static FMO model. Similarly, we can derive the pricing problem for these two aperture-based stochastic models. Note that the Karush-Kuhn-Tucker (KKT) optimality conditions are still necessary and sufficient for optimality because of the convexity of the objective function and the linearity of the constraints for the two models. In our two stochastic models, we check the target dose for each scenario with additional constraints (3 1) and (3 4), respectively, therefore we need to denote new dual multipliers associated with them to derive the KKT conditions. We first derive the pricing problem for the first model: let us denote the dual multipliers associated with constraints (3 1), (3 2)and (3 3) by π s j (j V T, s = 1,..., S), ω j (j V C ) and ρ k (k K), then the KKT conditions can be written as follows: πj s = 1 F (z s ) S zj s ω j = F (z) z j ρ k = S Dkjπ s j s + s=1 j V T for j V T, s = 1,..., S for j V C j V C Dkj ω j for k K 65

66 z s j = k K D s kjy k for j V T, s = 1,..., S z j = k K D kj y k for j V C y k ρ k = 0 for k K y k, ρ k 0 for k K. Similar with previous chapter, any solution to this system can be characterized by y 0 only: this vector then determines z s, z, π s, ω, and ρ. Let (ŷ; ˆπ s, ˆω, ˆρ) be an optimal pair of primal and dual solutions to a subproblem problem in which only apertures in the set ˆK K are considered, where we have set ŷ k = 0 for k K\ ˆK. Using the same method in Chapter 2, we conclude that the current solution is optimal if and only if the optimal solution to the following optimization problem ( S minimize k Kb Dij s ˆπ j s + s=1 j V T i A k j V C Dij ˆω j ) exceeds the threshold value ( ɛ S Dij s ˆπ j s + 1 ɛ s=1 j V T i N bk j V C Dij ˆω j ). (A) In the second stochastic model let us denote the dual multipliers associated with constraints (3 4), (3 5)and (3 6) by π s j (j V T, s = 1,..., S), ω j (j V ) and ρ k (k K), then the KKT conditions can be written as follows: πj s = 1 F (z s ) S zj s ω j = F + (z) z j S ρ k = Dkjπ s j s + s=1 j V T j V D kj ω j for j V T, s = 1,..., S for j V for k K 66

67 z s j = k K D s kjy k for j V T, s = 1,..., S z j = k K D kj y k for j V y k ρ k = 0 for k K y k, ρ k 0 for k K. Finally we conclude that the current solution is optimal if and only if the optimal solution to the following optimization problem ( S minimize k Kb Dij s ˆπ j s + s=1 j V T j V i A k D ij ˆω j ) exceeds the threshold value ( ɛ S Dij s ˆπ j s + 1 ɛ s=1 j V T j V i N bk D ij ˆω j ). (B) Note that we obtained the same form of expressions (A) and (B) as in Chapter 2: S s=1 j V T Dij s ˆπ j s + j V Dij C ˆω j and S s=1 j V T Dij s ˆπ j s + D j V ij ˆω j measure the per-unit change in objective function value if the intensity of beamlet i is increased for the two models, respectively Results In Chapter 2, by testing on clinical head-and-neck cancer cases, we showed the efficacy of this approach compared to the traditional method. The results indicate that delivery efficiency is very insensitive to the addition of traditional MLC constraints; however, jaws-only treatment requires about a doubling in beam-on time and number of apertures used. We also showed the importance of accounting for transmission effects. In this chapter, we focus on the effects of these two stochastic models where the transmission effects are taken into account. 67

68 In Chapter 2, we developed convergence and clinical stopping rules based on the treatment quality (in particular, we checked the DVHs in each iteration). Note the traditional DVH criteria for PTV are not suitable for the target in our stochastic models, we therefore check the target dose coverage (by estimating the probability that a fraction β of the target receives some certain dose, for different values of β) in each iteration in the convergence stopping rule. Our new two stopping rules are: Convergence: This stopping rule is based on observing that treatment plan quality, with respect to a particular criterion, has not improved markedly in recent iterations. In particular, in each iteration, we check the probabilities that at least 95%, 97% and 99% of target receive the prescription dose; the probabilities that at most 10% of target receive dose at hotspot; the average volume of bladder received doses at 65 Gy, 70 Gy, 75 Gy and 80 Gy over all scenarios; the average volume of rectum received doses at 60 Gy, 65 Gy, 70 Gy and 75 Gy over all scenarios. More formally, we say that we are satisfied with the solution with respect to a particular criterion if, in the last 5 iterations, the range of observed criterion values spans less than δ = 2%. Clinical: This stopping rule is based on observing that the treatment plan performance with respect to a clinical criterion has been satisfactory in the last 5 iterations. More formally, we say that we are satisfied with the solution with respect to a particular criterion if either the first stopping rule is satisfied or, in the last 5 iterations, for critical structures, the clinical DVH criteria are satisfied based on the average doses over all scenarios; for target, the average volume of target received prescription dose over all scenarios is more than 95% and the average volume of target received dose at hotspot over all scenarios is less than 10%. Tables 3-6 to 3-9 show the number of apertures and beam-on time for the two stochastic models according to four different MLC constraints, respectively. No big differences are observed with these two stochastic models. Taking Case 1 as an example, Figure 3-14 shows the target dose coverage for two stochastic models with two stopping rules. By observing all five cases, we conclude that in general, the second stochastic model yields slightly better target dose coverage. 68

69 3.4 Concluding Remarks Our stochastic optimization models take individual dose distributions for target into account and robust treatment plans can be obtained with only a modest number of scenarios. The target dose coverage improves not only for the scenarios included in the model but also in other scenarios. In general, the treatment plans obtained by our approach are better than that of the traditional method. The CPU time required for solving the beamlet-based models ranges about 3 7 minutes and for solving the aperture-based models ranges about 3 20 minutes for the 100-scenario models, and therefore feasible in a clinical setting. In addition, the second stochastic model obtained slightly better target dose coverage than the the first one while the CPU time required is relatively low (3-6 minutes for the beamlet-based model and 3-12 minutes for the aperture-based model). As a by product, we conclude that the traditional model overestimates the dose distribution to the target. 69

70 Table 3-6. C1: Number of apertures and beam-on time. Number of apertures Beam-on time Stochastic model 1 Stochastic model 2 Stochastic model 1 Stochastic model 2 case Clinical Convergence Clinical Convergence Clinical Convergence Clinical Convergence Average Table 3-7. C2: Number of apertures and beam-on time. Number of apertures Beam-on time Stochastic model 1 Stochastic model 2 Stochastic model 1 Stochastic model 2 case Clinical Convergence Clinical Convergence Clinical Convergence Clinical Convergence Average Table 3-8. C3: Number of apertures and beam-on time. Number of apertures Beam-on time Stochastic model 1 Stochastic model 2 Stochastic model 1 Stochastic model 2 case Clinical Convergence Clinical Convergence Clinical Convergence Clinical Convergence Average Table 3-9. C4: Number of apertures and beam-on time. Number of apertures Beam-on time Stochastic model 1 Stochastic model 2 Stochastic model 1 Stochastic model 2 case Clinical Convergence Clinical Convergence Clinical Convergence Clinical Convergence Average

71 Figure Target dose coverage for two models with two stopping rules. 71