A Fully-Automated Intensity-Modulated Radiation Therapy Planning System

A Fully-Automated Intensity-Modulated Radiation Therapy Planning System Shabbir Ahmed, Ozan Gozbasi, Martin Savelsbergh Georgia Institute of Technology Ian Crocker, Tim Fox, Eduard Schreibmann Emory University Abstract We designed and implemented a linear programming based IMRT treatment plan generation technology that effectively and efficiently optimizes beam geometry as well as beam intensities. The core of the technology is a fluence map optimization model that approximates partial dose volume constraints using conditional value-at-risk constraints. Conditional value-at-risk constraints require careful tuning of their parameters to achieve desirable plan quality and therefore we developed an automated search strategy for parameter tuning. Another novel feature of our fluence map optimization model is the use of virtual critical structures to control coverage and conformity. Finally, beam angle selection has been integrated with fluence map optimization. The beam angle selection scheme employs a bi-criteria weighting of beam angle geometries and a selection mechanism to choose from among the set of non-dominated geometries. The technology is fully automated and generates several high-quality treatment plans satisfying dose prescription requirements in a single invocation and without human guidance. The technology has been tested on various real patient cases with uniform success. Solution times are an order of magnitude faster than what is possible with currently available commercial planning systems and the quality of the generated treatment plans is at least as good as the quality of the plans produced with existing technology. 1 Introduction External beam irradiation is used to treat over 500,000 cancer patients annually in the United States. External beam radiation therapy uses multiple beams of radiation from different directions to cross-fire at a cancerous tumor volume. Thus, radiation exposure of normal tissue is kept at low levels, while the desired dose to the tumor volume is delivered. The key to the effectiveness of radiation therapy for the treatment of cancer lies in the fact that the repair mechanisms for 1

cancerous cells are less efficient than those of normal cells. Intensity-modulated radiation therapy (IMRT) is a relatively recent advance in treatment delivery in radiation therapy. In IMRT, the beam intensity is varied across each treatment field. Rather than being treated with a large uniform beam, the patient is treated instead with many small pencil beams (referred to as beamlets), each of which can have a different intensity. For many types of cancer, such as prostate cancer and head and neck cancer, the use of intensity modulation allows more concentrated treatment of the tumor volume, while limiting the radiation dose to adjacent healthy tissue (see Veldeman et al. [20] for comparisons of IMRT and non-imrt treatments on different tumor sites). IMRT treatment planning is concerned with selecting a beam geometry and beamlet intensities to produce the best dose distribution. Because of the many possible beam geometries, the large number of beamlets, and the range of beamlet intensities, there is an infinite number of treatment plans, and consistently and efficiently generating high-quality treatment plans is beyond human capability. Consequently, it is necessary to design and implement optimization-based decision support systems that can construct and suggest high-quality treatment plans in a short period of time. That is the goal of our research and the subject of this paper. Constructing an IMRT treatment plan that radiates the cancerous tumor volume without impacting normal structures is virtually impossible. Therefore, trade-offs have to be made. Radiation oncologists have developed a variety of measures to evaluate these trade-offs and to assess the quality of a treatment plan, such as the coverage of a target volume by a prescription dose, the conformity of a prescription dose around a target volume, the highest and the lowest doses received by a target volume. Furthermore, radiation oncologists review dose-volume histograms (DVHs) depicting the dose distributions at various structures (both target volumes and organs at risk) of treatment plans. Typically, a radiation oncologist specifies a set of dose-related minimum requirements (a set of prescriptions) that have to be satisfied in any acceptable treatment plan. The measures and DVHs are then used to choose among acceptable treatment plans. Further complicating the evaluation of treatment plans is the fact that the minimum requirements and their relative importance are subjective as are the underlying trade-offs they are trying to capture. As a result, most existing IMRT treatment planning systems are iterative in nature and necessitate human evaluation and guidance throughout the treatment plan construction process. This makes the process time-consuming and costly. One of the primary goals of our research is the development of fully-automated treatment plan generation technology that completely eliminates human intervention, thereby saving valuable time for the radiation oncologist and physicist. 2

Natural models of dose-volume requirements result in nonlinear and/or mixed integer programs, which are notoriously hard to solve. Recently, Romeijn et al. ([18]) observed that conditional value-at-risk constraints (C-VaR constraints), which are popular and commonly used in financial engineering, can be used to approximate dose-volume constraints. The use of C-VaR constraints has significant computational advantages as they can be handled using linear programming models, which are easily and efficiently solved. However, the parameters controlling the C-VaR constraints have to be chosen carefully to get an accurate approximation of the dose-volume constraints. Our treatment plan generation technology, which is based on an optimization model using C-VaR constraints, embeds an effective and efficient parameter search scheme to ensure high-quality approximations of the dose-volume constraints. Another innovation in our treatment plan generation technology is the use of virtual critical structures. Virtual critical structures surround target volumes and are implemented to control the dose deposits specifically at the boundary of the target volume. Our treatment plan generation technology incorporates an effective and efficient scheme for the selection of beam geometries, which is ignored in many commercial planning systems as it adds another layer of complexity to the search for high-quality treatment plans. Most commercial planning systems consider only beam geometries involving a small number of equi-distant beam angles around a circle (on the order of 5 to 8). Because of the computational efficiency of our core plan generation technology, we are able to embed a scheme that optimizes the beam geometry. At the heart of the scheme is a multi-attribute beam scoring mechanism based on a treatment plan constructed for a beam geometry involving a large number of beam angles (on the order of to 24). In summary, we have developed treatment plan generation technology that optimizes both beam geometry and beamlet intensities in a short amount of time. The technology is fully automated and generates several high-quality treatment plans satisfying the provided minimum requirements in a single invocation and without human guidance. The technology has been tested on various real patient cases with great success. Solution times range from a few minutes to a quarter of an hour, which is an order of magnitude faster than what is possible with currently available commercial planning systems. Furthermore, the quality of the generated treatment plans is at least as good as those produced with existing technology. The remainder of the paper is organized as follows. In Section 2, we introduce IMRT treatment plan generation and evaluation in more detail. In Section 3, we describe the core components of the IMRT treatment plan generation technology that we have developed. Finally, in Section 4, we 3

present the results of an extensive computational study. 2 Problem Description Intensity-modulated radiation therapy (IMRT) is an advanced mode of high-precision radiotherapy that utilizes computer-controlled mega-voltage x-ray accelerators to deliver precise radiation doses to a cancerous tumor or specific areas within the tumor. The radiation dose is designed to conform to the three-dimensional (3-D) shape of the tumor by modulating or controlling - the intensity of the radiation beam to focus a higher radiation dose to the tumor while minimizing radiation exposure to surrounding normal tissues. Typically, combinations of several intensity-modulated fields coming from different beam directions produce a custom tailored radiation dose that maximizes tumor dose while also protecting adjacent normal tissues. IMRT is being used to treat cancers of the prostate, head and neck, breast, thyroid, lung, liver, brain tumors, lymphomas and sarcomas. Radiation therapy, including IMRT, stops cancer cells from dividing and growing, thus slowing tumor growth. In many cases, radiation therapy is capable of killing cancer cells, thus shrinking or eliminating tumors. For a more detailed description, review, history, and physical basis of IMRT, see Bortfeld ([3]), Boyer et al. ([4]), Shepard ([19]), and Webb ([21]). An IMRT treatment plan has to specify a beam geometry (a set of beam directions) and for each beam direction a fluence map (a set of beamlet intensities; a beam can be thought of as consisting of a number of beamlets that can be controlled individually). Typically, a small number of equi-spaced coplanar beam directions are used. Coplanar beam directions are obtained by rotating the gantry while keeping the treatment couch in a fixed position parallel to the gantry axis of rotation. There are practical reasons for limiting the number of beam directions (between 5 and 8) as it reduces patient positioning times, chances for patient positioning errors, and delivery time, but considering only a restricted set of beam geometries is driven primarily by the computational complexity it introduces in the treatment planning process. For modeling purposes, the structures (target and critical structures) are discretized into cubes called voxels (e.g., cubes of 5 5 5 mm) and the dose delivered to each voxel per intensity of each beamlet is calculated (see Ahnesjo [1], Mackie et al. [12], and Mohan et al. [13] for dose calculation methods). The total dose received by a voxel is assumed to be the sum of doses deposited from each beamlet. The goal is to build a treatment plan that creates a dose distribution that will destroy, or at least damage, target cells while sparing healthy tissue by choosing proper beam directions and 4

beamlet intensities. In order to guide the construction of a treatment plan, a radiation oncologist specifies a set of requirements that have to be satisfied in any acceptable treatment plan. These requirements are in the form of a minimum prescription dose for target structure voxels and a maximum tolerance dose for nearby critical structure voxels. A prescription dose is the dose level necessary to destroy or damage target cells, while a tolerance dose is the dose level above which complications for healthy tissues may occur. The requirements are grouped into two types: (1) full-volume constraints and (2) partial-volume constraints. Full-volume constraints specify dose limits that have to be satisfied by all voxels of a structure. Partial-volume constraints, on the other hand, specify dose limits that have to be satisfied by a specified fraction of the voxels of a structure. The choice of a full-volume or a partial-volume constraint for a healthy structure depends on the functionality of the organ. For example, a partial-volume constraint is used when the organ at risk will not be damaged as long as a specified volume receives a dose below the tolerance dose. In addition to the set of dose requirements defining acceptable treatment plans, clinical oncologists use a set of evaluation metrics to assess the quality of a treatment plan. The cold spot and hot spot metrics are evaluation measures that consider the minimum and maximum dose received by a voxel in a structure relative to the prescription dose. The coverage metric considers the fraction of voxels in the target structure receiving a dose greater than the prescription dose. The conformity metric considers the ratio of the number of voxels in the target region and surrounding tissue receiving a dose greater than the prescription dose to the number of voxels in the target structure itself receiving a dose greater than the prescription dose. More formally, these evaluation metrics are defined as follows. Let N denote the set of beamlets, S the set of critical structures, T the set of target structures, V s the number of voxels in structure s, and z js the dose received by voxel j of structure s. (Of course z js depends on the treatment plan.) The cold spot of target structure s T is defined as the ratio of the minimum dose received by any of the voxels of the structure to the prescription dose for that structure, i.e. cold spot(s) = min{z js : j = 1,..., V s } P D s, (1) where P D s is the prescription dose for target structure s. Similarly, the hot spot of target structure s T is defined as the ratio of the maximum dose received by any of the voxels of the structure to 5

the prescription dose, i.e. hot spot(s) = max{z js : j = 1,..., V s } P D s. (2) Ideally, every voxel in a target structure receives exactly the prescription dose. The cold spot metric and hot spot metric measure the deviation from this ideal situation by examining the voxel receiving the smallest dose and the voxel receiving the largest dose. The coverage of target structure s T is the proportion of voxels receiving a dose greater than or equal to the prescription dose P D s, i.e. coverage(s) = V s I + (z js P D s ) V s, (3) where I + is the indicator function for the non-negative real line, i.e., I + (a) is equal to 1 if a 0 and 0 otherwise. For simplicity assume that T = {s}. Note that 0 coverage(s) 1 and that values closer to 1 are preferable. The conformity of target structure s is the ratio of the number of voxels in the structure and its surrounding tissue receiving a dose greater than or equal to the prescription dose P D s to the number of voxels in the structure itself receiving greater than or equal to the prescription dose, i.e., conformity(s) = V s s S T V s I + (z js P D s ) I + (z js P D s ) = 1 + V s I + (z js P D s ) s S V s I + (z js P D s ). (4) Note that 1 conformity(s) and that values closer to 1 are preferable. The coverage and conformity metrics are illustrated in Figure 1. In the schematic on the left, the set of voxels receiving a dose greater than or equal to the prescription dose is exactly the set of voxels in target structure s and thus coverage(s) = conformity(s) = 1; this is the ideal solution. In the schematic in the center the set of voxels receiving a dose greater than or equal to the prescription dose is twice the size of the set of voxels in target structure s (and includes all the voxels of s) and thus coverage(s) = 1 and conformity(s) = 2. Finally, in the schematic on the right, the set of voxels receiving a dose greater than or equal to the prescription dose is exactly half the set of voxels in target structure s and thus coverage(s) = 0.5 and conformity(s) = 1. Ideally, every voxel in a target structure receives exactly the prescription dose and all the voxels in the surrounding tissue (e.g., in the nearby critical structures) receive no dose at all. The coverage metric measures how well we cover the voxels in 6

TARGET TARGET TARGET Coverage = 1 Conformity = 1 Coverage = 1 Conformity = 2 Coverage = 0.5 Conformity = 1 Prescription Dose Region Figure 1: Coverage and conformity metrics the target structure. The conformity metric measures, in a sense, what happens at the boundary of the target structure. It measures the price that has to be paid, in terms of exposure of healthy tissue, to achieve high coverage. Due to the fact that coverage and conformity consider the target as a whole, these metrics are typically considered of higher importance than the cold spot and hot spot metrics. (See Lee et al. ([8, 9]) for earlier use of these metrics for plan evaluation.) As we will see in the next section, the evaluation metrics are optimized implicitly; the evaluation metrics are not incorporated explicitly in the optimization problem. 3 Methodology In this section, we describe the key components of the proposed IMRT treatment plan generation technology. The core of the technology is a fluence map optimization (FMO) model based on linear programming. This model is similar to the one proposed by Romeijn et al. ([18]) and approximates partial volume constraints using conditional value-at-risk (C-VaR) constraints. C- VaR constraints require careful tuning of their parameters to achieve desirable plan quality and therefore an automated search strategy for parameter tuning has been developed. (See Preciado- Walters et al. [14] and Lee et al. [8] for exact mixed integer programming formulations of partial volume constraints.) A novel feature of the fluence map optimization model is the use of virtual 7

critical structures to control coverage and conformity. Finally, beam angle selection has been integrated with fluence map optimization. The beam angle selection scheme employs a bi-criteria weighting of beam angle geometries and a selection mechanism to choose from among the set of non-dominated geometries. 3.1 Fluence Map Optimization Model Recall that N denotes the set of beamlets, S the set of critical structures, T the set of target structures, V s the number of voxels in structure s, and D ijs the dose received by voxel j of structure s per unit intensity of beamlet i. Let x i be the intensity of beamlet i, i.e., a decision variable, then the dose z js received by voxel j of structure s, is z js = i N D ijs x i j = 1,..., V s ; s S T. (5) Full volume constraints Let L s and U s be the prescribed lower and upper dose limits for structure s, respectively. The full-volume constraints for dose limits are L s z js U s j = 1,..., V s ; s S T. (6) Partial volume constraints The dose-volume constraints for target structures are formulated as 1 V s c s (c s z js ) + L α s s T. (7) This C-VaR constraint enforces that the average dose in the (1 α s )-fraction of voxels of target structure s receiving the lowest amount of dose is greater than or equal to L α s. When satisfied at least α s 100 per cent of the voxels will receive a dose greater than or equal to L α s. For example, with α s = 0.9 and L α s = 30 at least 90% of the voxels will receive a dose greater than 30 Gy. Here c s is a free variable associated with the dose-volume constraint of structure s. Similarly, the partial-volume constraints for critical structures are 1 V s c s + (z js c s ) + Us α s S. (8) 8

For critical structure s the average dose in the subset of size (1 α s ) receiving the highest amount of dose is required to be less than or equal to U α s. When satisfied at least α s percent of the voxels will receive a dose less than or equal to U α s. Cold spot and hot spot requirements As mentioned before the cold spot metric (1) for a target structure s T is defined as the ratio of the minimum dose received by the voxels of the target structure to the prescription dose, and the hot spot metric (2) for a target structure s T is defined as the ratio of the maximum dose received by the voxels of the target structure to the prescription dose. Prescribed requirements on cold spot and hot spot can be enforced by the bounds L s and U s in the full volume constraints (6). Coverage requirements The coverage metric (3) of a target structure s T is the proportion of voxels receiving a dose greater than or equal to the prescription dose P D s. Due to the indicator function in the expression (3), requirements on coverage are hard to model using convex constraints. Instead, we use partial volume constraints to enforce the coverage requirements using the following result. Proposition 1 Given α s (0, 1), if there exists a c s such that 1 V s c s (c s z js ) + P D s (9) then coverage(s) α s. (10) A proof of this proposition can be found in the appendix. Thus coverage(s) can be increased by increasing α s in the partial volume constraints (7) with L α s = P D s. Of course α s = 1 is most preferable but such an aggressive value may lead to problem infeasibility. Note that in this case (i.e. when α s = 1) the partial volume constraint reduces to a full volume constraint. Conformity requirements The conformity metric (4) of a target structure s T is the ratio of the total number of voxels in the body receiving a dose greater than or equal to the target prescription dose P D s to the number of voxels in the target receiving greater than or equal to prescription dose in the target structure. 9

Unlike cold spot, hot spot and coverage; conformity requirements cannot be directly enforced by full or partial volume constraints on the dose on the target structure. This is because the conformity metric needs to consider dose deposited outside the target. In order to keep track of the intended dose for the target s that is deposited outside the target, we introduce a virtual critical structure around the target structure s. This is typically a 30 mm band around the target structure s, denoted by b(s), which is included in S. Price et al. ([15]) and Lee et al. ([9]) showed on clinical cases that limiting or minimizing the dose on virtual critical structures around the target structures can improve the conformity of the plans. Girinsky et al. ([6]) also used virtual critical structures to avoid high doses around the tumor volume. Assuming that all of the dose intended for the target s T will be deposited in the voxels in s and in b(s), the only structures in S that can have more than the prescribed dose P D s will be b(s). Then from (4) conformity(s) = 1 + V b(s) V s I + (z js P D s ) I + (z js P D s ) = 1 + coverage(b(s)) V b(s) coverage(s) V s, where the second equality follows from the definition of coverage. We know that by enforcing the partial volume constraint (7) on s with L α s = P D s and given α s (0, 1) we ensure coverage(s) α s. Similarly, it can be shown that by enforcing the partial volume constraint (8) on b(s) with U α b(s) = P D s and given α b(s) (0, 1), we ensure coverage(b(s)) (1 α b(s) ). Thus conformity(s) 1 + (1 α b(s)) V b(s) α s V s. (11) Therefore we can reduce conformity(s) by increasing α s as well as by increasing α b(s). Setting these parameters at their maximum value of 1 typically leads to problem infeasibility, and so a careful selection has to be made. This is further discussed in Section 3.2. 10

The LP model The final formulation of our FMO model is min 1 V s s S V s z js s T 1 V s V s z js (12) s.t. z js = i N D ijs x i j = 1,..., V s ; s S T (13) L s z js U s j = 1,..., V s ; s S T (14) 1 V s c s + (z js c s ) + Us α s S (15) (1 α s )V s 1 V s c s (c s z js ) + L α s s T (16) (1 α s )V s x i 0 i = 1,..., N (17) z js 0 j = 1,..., V s ; s S T (18) c s free s S T. (19) The objective function (12) attempts to decrease the average dose on the critical structures while increasing the average dose on the target structures. (See Kessler et al. [7] and Yang and Xing [23] for objective functions used in IMRT formulations). Constraint (13) calculates the total dose z js deposited on to voxel j of structure s from all beamlets. Constraints (14) are the full volume constraints on the structures according to the prescribed dose limits and the cold spot and hot spot considerations. Constraints (15) are the partial volume constraints on the critical structures. Since S includes the virtual critical structure b(s) for each target s T, these constraints serve to enforce conformity requirements. Constraints (16) are the partial volume constraints on the target structures and enforce coverage requirements. Finally, the beamlet intensities and total dose deposited are required to be non-negative while the artificial variables c s are free. Note that the nonlinear function ( ) + in the partial volume constraint can be linearized in the standard way leading to a linear programming formulation of the above FMO model. 3.2 Parameter Search As mentioned before, the coverage and conformity corresponding to a target structure s T in a treatment plan resulting from the FMO model (12)-(19) depends on the values of the parameters α b(s) and α s in the upper and lower partial volume constraints (15) and (16) for b(s) and s T, respectively. In this section, we describe a search technique for appropriate values of these 11

parameters. For simplicity of exposition we assume that there is only one target structure, i.e., T = {t}. Recall that, for good coverage and conformity for the target t, high values of α b(t) and α t are preferable, however this may lead to problem infeasibility. Let F t = {(α b(t), α t ) (0, 1) 2 : The FMO model (12)-(19) is feasible}. Note that if (α t, α b(t) ) F t, α t α t and α b(t) α b(t), then (α t, α b(t) ) F t. Assuming F t is non-empty we would like to maximize α t and α b(t) over F t. Such solutions will lie on the upper boundary of F t. Our search technique starts with an initial solution and goes through several search phases to identify solutions on the upper boundary of F t. Figure 2 illustrates the search phases, which are now detailed. Initial values of the parameters α t and α b(t) are chosen based on a minimum coverage requirement M incov and a maximum allowed conformity value M axconf. In particular α 0 t = MinCov β (20) α 0 b(t) = (1 MinCov(MaxConf 1)V t V b(t) ) β (21) where 0 < β < 1 is scaling factor to account for the fact that the bounds (10) and (11) are conservative. In our implementation for case studies we used β = 0.9. We solve the FMO model (12)-(19) with the initial parameter values (α 0 b(t), α0 t ). If the problem is infeasible, i.e. (α 0 b(t), α0 t ) F t, then we execute Phase 0. Here we decrease both parameters by λ, where 0 < λ < 1, and resolve the FMO model. This is continued until a feasible set of parameters is found. The structure of the set F t guarantees that if F t is non-empty, then Phase 0 will produce a feasible set of parameters. Once we have initial feasible parameters, we execute Phase 1, where we increase both α b(t) and α t by λ and resolve the FMO model. This is repeated until we reach an infeasible set of parameters. The last set of feasible parameters, denoted by (α 1 b(t), α1 t ), then is a solution near the upper boundary of F t. Next we execute Phase 2 where, starting from (α 1 b(t), α1 t ), we increase α 1 t by λ until we reach an infeasible set of parameters. The motivation here is that, since increasing α t improves both coverage and conformity, we want to obtain additional solutions with good coverage and conformity values. The last feasible solution from Phase 2 is denoted by (α 2 b(t), α2 t ). In order to find additional candidate solutions on the boundary of F t, we execute Phase 3. Here we first reduce α b(t) by λ, and then start Phase 2 to increase α t. Note that in the solutions produced 12

a t (Until Infeasibility) F t (a b(t), a t ) (a b(t) 3, a t 3 ) Final Values (Until Infeasibility) Phase 3 Phase 2 (a b(t) 2, a t 2 ) (Until Infeasibility) 0 0 (a b(t), a t ) Initial Values Phase 0 Phase 1 (a b(t) 1, a t 1 ) Phase 4 F t (a b(t), a t ) (Until Infeasibility) (Until a Feasible Solution) a b(t) Figure 2: Parameter Search for α t and α b(t) 13

in this phase the conformity values may be worse than those from Phase 2, however coverage will not worsen. Finally, if Phases 2 and 3 are not able to produce feasible parameters that are greater than (αb(t) 1, α1 t ), then we execute Phase 4. Here we increase α b(t) by λ in order to find additional solutions near the upper boundary of F t. The step size λ is an important component of the search process. From initial experimentations, we chose this to be λ = 0.01. 3.3 Beam Selection Careful selection of beam angles is an important part of creating IMRT plans as these angles provide the candidate beamlets for which the intensities are optimized. Holder et al. ([5]) provide a rigorous presentation of the beam selection problem and some comparisons of current algorithms on phantom cases. (See Webb [2] for a classification of beam selection algorithms proposed in the literature.) Clinical use of the beam selection algorithms are rare and most commercial planning systems still rely on manual selection of beam angles by planners (Pugachev and Xing [17]). In this section, we describe a scheme for selecting a set of beam angles (geometry/configuration) to deliver the treatment from. Suppose there are 18 candidate angles and we wish to select 8 delivery angles, then there are 43,758 choices making complete enumeration clinically impossible. (See Liu et al. [11] for a complete enumeration based method.) Instead of evaluating all possible combinations using fluence optimization, we use a bi-criteria scoring approach to limit our choices to a relatively small number of non-dominated combinations. We compare our beam selection methodology with scoring-based approach of Pugachev and Xing ([16]) in Section 4. The first step is to use the FMO model and parameter search to determine a treatment plan satisfying prescription criteria assuming that all candidate beam angles (say there are M of these) can be used. Let x i be the intensity of beamlet i in beam (angle) B (where B = 1,..., M) and z jt be the dose delivered to voxel j of the target structure t (PTV) in this treatment plan. Each beam (angle) B is then assigned the following two weights: Total Dose Delivered to PTV. The total dose delivered to PTV by beam B is computed as DP T V B = i B V t D ijt x i. (22) Preferring beams with a higher value of DP T V B favors beams that deliver higher dose to the target. The disadvantage of this measure is that it does not directly consider the dose 14

received by PTV voxels in the lower tail of the dose distribution, i.e., cold spots, and cold spots have a significant effect on treatment quality. Weighted Sum of Dose Delivered to the Cold Region of PTV. We define the cold region C t of PTV as the set of voxels which receive a dose less than 10% above the minimum dose observed in PTV, i.e., C t = {j = 1,..., V t : z jt 1.10 min j (z jt)}. (23) The weighted sum of dose delivered to the cold region of PTV from beam angle B is computed as W P T V B = D ijt x 1 i max{z (24) i B j C t jt, ɛ}, where ɛ > 0 is included to avoid division by zero. Preferring beams with higher value of W P T V B favors beams that deliver more dose to the cold region of the target. Note that we distinguish voxels in the cold region by assigning weights inversely proportional to the actual dose received by these voxels. Thus, a beam delivering the same amount of dose as another beam but to colder voxels will have a higher score. The advantage of this measure is that it can help to identify beams, the blocking of which may result in a significant loss of dose delivery to the cold region of PTV and thus lower tumor control. However, a scoring mechanism purely based on W P T V B values may lead to formation of new cold spots, because these beams are likely to intersect at the same part of PTV and may be ineffective in other parts of PTV. Next, we consider each possible choice of L delivery angles from the total M angles. (Note there are ( M L ) possibilities.) For each of the possible configurations, i.e., a set of beam angles C {1,..., M} such that C = L, we compute the configuration weights, i.e., DP T V (C) = B C DP T V B, and W P T V (C) = B C W P T V B. Next, we identify a set K of non-dominated configurations, i.e., K = {C : D s.t. DP T V (D) > DP T V (C) and W P T V (D) > W P T V (C)}. Ideally, to choose between the non-dominated configurations in K, we would generate an optimized treatment plan for each (using the FMO model with parameter search) and select the one with the 15

highest quality. However this may be too time consuming. Instead, we perform a greedy search. We start with an arbitrary configuration C K and execute Phase 1 and Phase 2 of the parameter search. Let the resulting set of parameter values be (αb(t) 2, α2 t ). Next, we search for a configuration Ĉ K for which (αb(t) 2, α2 t + λ) is feasible. If no such configuration Ĉ exists, then configuration C is selected. On the other hand, if such a configuration Ĉ exists, we iterate and execute Phase 1 and Phase 2 of the parameter search for Ĉ starting from (α2 b(t), α2 t ). 4 Computational Study In this section, we present the results of a computational study conducted using three real-life cases: a pediatric brain case, a head and neck case, and a prostate case. The computational study focuses on the value of the techniques and solution schemes introduced above: (1) the use of virtual critical structures to construct high-quality treatment plans in terms of coverage and conformity, (2) the development of an automated, systematic parameter search scheme for treatment plan construction models using conditional value at risk constraints to model partial dose volume constraints, and (3) the development of an efficient and effective beam angle selection scheme. 4.1 Case Descriptions The computational study is conducted using three real-life cases: a pediatric brain case, a headand-neck case, and a prostate case. As these cases represent different parts of the body, and thus a variety of challenges in terms of treatment plan generation, their use will demonstrate the robustness of our approach; no tuning is necessary for the specific cases. Information concerning the prescription requirements for the tumor as well as for the critical structures (organs at risk) in terms of full volume and partial volume dose constraints for the three cases can be found in Tables 1, 2, and 3. Partial dose volume constraints for the tumor (PTV) and the virtual critical structure (VCS) are computed using the formulas presented in Section 3 using a minimum coverage requirement of 0.95 and a maximum conformity requirement of 1.2. Observe that there are no partial dose-volume constraints for critical structures in the pediatric brain case. To be able to analyze and judge the value of the different techniques and solution schemes we have developed, we start with base settings in each of the three cases. With base settings radiation is delivered from 8 equi-spaced beams in the pediatric brain case, from 8 equi-spaced beams in the head and neck case, and from 6 equi-spaced beams in the prostate case. Furthermore, the full and 16

Constraint Type Structure #Voxels Percentage LB (cgy) UB (cgy) PTV 1,620-0 3,300 Hypothalamus 88-0 2,160 Right cochlea 26-0 1,260 Full volume Left cochlea 21-0 1,260 Pituitary 11-0 2,160 Left eye 158-0 500 Right eye 163-0 500 VCS 6,581-0 3,300 Partial volume PTV 1,620 95 3,060 - VCS 6,581 95-3,060 Table 1: Brain Case - Structures and Constraints partial dose volume constraints as specified in Tables 1, 2, and 3 are used except for those related to the virtual critical structure. 4.2 Value of the Virtual Critical Structure The virtual critical structure was introduced to improve conformity. Our first computational experiment focuses on whether the introduction of the virtual critical structure indeed improves conformity and, if so, by how much. This simply means that we compare the base setting with and without the full and partial dose volume constraints related to the virtual critical structure. The results are presented in Table 4. We see that the effect of introducing the virtual critical structure is dramatic. For all three cases, the conformity is significantly improved. We also observe that the coverage is improved in all three cases, although only slightly. We pay a small price in terms of the coldspot; for both the head and neck case and the prostate case, the minimum dose has decreased slightly. 4.3 Value of the Parameter Search As the treatment plan construction model only indirectly attempts to optimize coverage and conformity, we may be able to improve coverage and conformity by judiciously adjusting the parameters α P T V and α V CS. In Section 3.2, we have outlined an automated, systematic parameter search 17

Constraint Type Structure #Voxels Percentage LB (cgy) UB (cgy) PTV 2,647-0 6,120 Brainstem 1,075-0 5,400 Spinal cord 387-0 4,500 Globe LT 388-0 2,000 Globe RT 355-0 2,000 Full volume Optic chiasm 13-0 5,400 Optic nerve LT 26-0 5,400 Optic nerve RT 10-0 5,400 Parotid LT 811-0 5,400 Parotid RT 832-0 5,400 VCS 7,634-0 6,120 PTV 2,647 95 5,100 - Partial volume Parotid LT 811 50-2,600 Parotid RT 832 50-2,600 VCS 7,634 93-5,100 Table 2: Head & Neck Case - Structures and Constraints scheme to produce treatment plans with better coverage and conformity than the plans produced with the initial parameter values. In the next computational experiment, we focus on the improvement in coverage and conformity that can be achieved through the parameter search. We compare the base settings (with full and partial dose volume constraints for the virtual critical structure included) with and without parameter search. The results are presented in Table 5. We observe that there are significant improvements for coverage and conformity for the brain and the prostate case. However, for the head-and-neck case the slight improvement in coverage is offset by a deterioration in conformity. The parameter search for the brain case and for the head and neck case end with a Phase 3 iteration, in which the virtual critical structure constraint is relaxed to see if that allows for treatment plans with higher coverage. As a result, the value of conformity increases slightly in the last iteration. More details are provided in Figure 3 where we plot the change in the evaluation metrics during the course of the parameter search. 18

Constraint Type Structure #Voxels Percentage LB (cgy) UB (cgy) PTV 239-0 3,300 Full volume Rectum 1,267-0 2,160 Bladder 1,513-0 1,260 VCS 2,544-0 3,300 PTV 239 95 7,560 - Rectum 2,544 70-7,560 Partial volume Rectum 2,544 50-4,500 Bladder 1,513 50-4,500 VCS 2,544 98-7,560 Table 3: Prostate Case - Structures and Constraints Brain Head & Neck Prostate without VCS with VCS without VCS with VCS without VCS with VCS Coverage 0.954 0.957 0.953 0.956 0.933 0.950 Conformity 1.533 1.274 1.742 1.230 1.601 1.264 Coldspot 0.669 0.704 0.328 0.311 0.853 0.829 Hotspot 1.078 1.078 1.200 1.200 1.150 1.150 Table 4: Value of Virtual Critical Structure 4.4 Beam Angle Selection Carefully selecting beam angles may significantly enhance the ability to construct high-quality treatment plans. However, due to the large number of possible beam angles and the complexity of generating high-quality treatment plans given a set of beam angles, beam angle selection has not been extensively investigated and few effective schemes for beam angle selection exist. In the next computational experiments, we focus on the effect of the beam angle selection scheme we have developed. A core ingredient of our scheme is identifying non-dominated beam configurations with respect to two quantities computed for a treatment plan in which all candidate beam angles are available for use: (1) the dose delivered to the PTV and (2) the dose delivered to cold regions of the PTV. In Figures 4, 5, and 6, we plot the scores for all 8-beam configurations for the brain 19

Brain Head & Neck Prostate without with without with without with parameter parameter parameter parameter parameter parameter search search search search search search Coverage 0.957 0.984 0.956 0.959 0.950 1.000 Conformity 1.274 1.212 1.230 1.261 1.264 1.046 Coldspot 0.704 0.822 0.311 0.313 0.829 1.000 Hotspot 1.078 1.078 1.200 1.200 1.150 1.150 Table 5: Value of Parameter Search and head and neck cases and all 6-beam configurations for the prostate case selected from 18 equispaced candidate beams and highlight the configurations that are non-dominated. The figures show that few non-dominated configurations exist. In Table 6, we provide additional information to support the observation that very few non-dominated configurations exist. For configurations of different sizes, we report the total number of configurations as well as the number of non-dominated configurations. Next, we explore the effect of carefully selecting beam angles, and thus of integrating all the techniques and solution schemes developed. The results are presented in Table 7. The results are mostly self-explanatory. They clearly demonstrate the value of the various techniques and solution schemes we have introduced. However, these evaluation metrics only tell part of the story. Clinicians like to examine dose volume histograms as well. We show dose volume histograms in Figures 7, 8, and 9 for the brain case, the head and neck case, and the prostate case, respectively. Each of the figures displays dose volume histograms for three treatment plans: (1) a plan with equispaced beams obtained by solving the model, but without parameter search, (2) a plan with equispaced beams obtained by solving the model incorporating parameter search, and (3) a plan with optimized beam angles obtained by solving the model incorporating parameter search. The dose volume histograms show that improving coverage and conformity (i.e., by performing a parameter search) can have a negative impact on the dose volume histograms of the organs at risk (increased doses delivered), but that by carefully selecting the beam angles these undesirable effects can be mostly negated. This can be observed especially well in the head and neck case; consider for example the brainstem. 20

Configuration Size # Non-Dominated Configurations # Configurations Brain Head & Neck Prostate 2 4 2 2 153 3 5 3 2 816 4 7 4 3 3,060 5 8 5 3 8,568 6 10 5 2 18,564 7 13 7 2 31,824 8 12 8 3 43,758 9 12 7 3 48,620 Table 6: Number of Non-dominated Configurations for Various Configuration Sizes Comparison with another Beam Selection Algorithm To further validate our approach, we compare our results to the results obtained by selecting the beam angles according to the pseudo Beam s-eye-view (pbev) scores introduced by Pugachev and Xing ([16]). The steps for the pbev calculation of a given beam angle are as follows: 1. Find the voxels crossed by each beamlet of the beam angle; 2. Assign each beamlet an intensity that delivers a dose equal to or higher than the prescription dose in every target voxel; 3. For each OAR or normal tissue voxel crossed by the beamlet, calculate the factor by which the beamlet intensity has to be multiplied to ensure the tolerance dose is not exceeded (factor less than or equal to 1); 4. Find the minimum factor among the ones calculated in Step 3; 5. Adjust the beamlet intensities obtained in Step 2 using the minimum factor. The resulting values, referred to as the maximum beam intensity profile, represent the maximum usable intensity of the beamlet; 6. Perform a forward dose calculation using the maximum beam intensity profile; 21

Case Evaluation Equi-spaced Equi-spaced & Equi-spaced & Optimized angles & measure VCS VCS & VCS & Parameter search Parameter search Coverage 0.954 0.957 0.984 1.000 Brain Conformity 1.533 1.274 1.212 1.125 Coldspot 0.669 0.704 0.822 1.000 Hotspot 1.078 1.078 1.078 1.078 Coverage 0.953 0.956 0.959 0.974 Head & Conformity 1.742 1.230 1.261 1.232 Neck Coldspot 0.328 0.311 0.313 0.310 Hotspot 1.200 1.200 1.200 1.200 Coverage 0.933 0.950 1.000 1.000 Prostate Conformity 1.601 1.264 1.046 1.033 Coldspot 0.853 0.829 1.000 1.000 Hotspot 1.150 1.150 1.150 1.150 Table 7: Evaluation Metrics for Different Schemes 7. Compute the score for chosen beam angle i according to an empiric score function as follows: S i = 1 V t V t ( d ij P D t ) 2 (25) where d ij is the dose delivered to voxel j from beam angle i, V t is the number of voxels in the target, and P D t is the target prescription dose. We selected the final configuration simply by picking the highest scoring beams. (In [16], Pugachev and Xing manually pick the beams in the case examples when some high-scoring angles are too close to each other to create a separation.) We computed beam angle configurations for the cases using pbev scores and our proposed scores, and then performed fluence map optimization using our technology. Multiple solutions are produced for each case by varying the number of beam angles. The configuration sizes are 6, 7, 8 and 9 for both brain and head and neck cases, and 4, 5, 6 for the prostate case. The average values of the evaluation metrics (over the different configuration sizes) are presented in Table 8. The values of coverage and conformity are better in all cases with our proposed scheme. 22

Case Evaluation pbev Optimized Angles meatric Coverage 0.974 0.991 Brain Conformity 1.209 1.146 Coldspot 0.949 0.989 Hotspot 1.078 1.078 Coverage 0.966 0.971 Head & Conformity 1.219 1.214 Neck Coldspot 0.344 0.311 Hotspot 1.200 1.200 Coverage 1.000 1.000 Prostate Conformity 1.045 1.033 Coldspot 1.000 1.000 Hotspot 1.150 1.150 Table 8: Comparison of Evaluation Metrics for Beam Angles Selected with pbev and Our Scheme The improvement is significant especially for the brain case. The main difference between our scoring scheme and the pbev scoring scheme is that we score beams based on their exact contributions to the delivered doses to target structures in an optimized solution rather than just using their geometric properties. Another important difference is that we consider multiple beam configurations in the parameter search phase, which allows us to adjust dynamically in response to observed interaction effects between beams. In the pbev algorithm there is only one configuration which is chosen based on initial parameter values. 4.5 Beam Configuration Size The results presented so far assumed that the desired number of beam angles was decided in advance. We have seen that by carefully selecting beam angles substantially better treatment plans can be generated. This suggests that it may be possible to get high-quality treatment plans with fewer beam angles. Table 9 presents the values of the evaluation criteria for different sizes of beam configurations. The same information is displayed graphically in Figure 10. A number of observations can be made. As expected, there are diminishing returns when we 23

2 beams 3 beams 4 beams 5 beams 6 beams 7 beams 8 beams 9 beams Coverage 0.872 0.898 0.939 0.955 0.963 1.000 1.000 1.000 Brain Conformity 1.443 1.400 1.283 1.222 1.214 1.144 1.125 1.099 Coldspot 0.636 0.736 0.912 0.936 0.956 1.000 1.000 1.000 Hotspot 1.078 1.078 1.078 1.078 1.078 1.078 1.078 1.078 Coverage 0.923 0.937 0.961 0.965 0.968 0.971 0.974 0.973 Head & Neck Conformity 1.420 1.350 1.286 1.252 1.219 1.243 1.232 1.162 Coldspot 0.331 0.341 0.326 0.317 0.301 0.320 0.310 0.314 Hotspot 1.200 1.200 1.200 1.200 1.200 1.200 1.200 1.200 Coverage 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Prostate Conformity 2.515 1.556 1.029 1.038 1.033 1.042 1.042 1.038 Coldspot 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Hotspot 1.150 1.120 1.150 1.150 1.150 1.150 1.150 1.150 Table 9: Metric Values for Different Beam Configuration Sizes continue to increase the size of the configuration. (This is especially clear for the prostate case where treatment plans of almost equal quality are produced for configurations of size 4 and up.) Furthermore, treatment plans with good quality can already be achieved with configurations of relatively small size (which has many practical advantages). If we compare the results presented in Table 7 to those presented in Table 9, we see that carefully selecting beam angles allows us to construct treatment plans of equal or better quality than those that can be obtained with equispaced beams with fewer beams. The above results indicate that our beam angle selection scheme produces beam configurations that allow the construction of high-quality treatment plans. However, it is possible that other beam configurations may allow even better treatment plans. Therefore, to further assess the quality of the beam angle configurations produced by our scheme, we compare the beam configurations produced by our scheme to those produced by an optimization approach based on an integer programming formulation (Lee et al. [8, 9], Yang et al. [22], and Lim et al. [10]). Due to the computational requirements of the optimization approach this comparison can only be performed for settings with a relatively small number of candidate beam angles. We experimented with selecting 5 beam angles out of 8 equi-spaced candidate beam angles for the brain and the head and neck case. Both times, our selection scheme produced the optimal configuration, i.e., the same configuration that the optimization approach produced. It is informative to look at the difference in required computation time; see Table 10. It is clear that the optimization approach will become computationally prohibitive when 10 or more candidate beam angles are used. 24

Time (secs.) Ratio Scheme MIP Brain 348 5,676 1 : 16 Head & Neck 363 16,988 1 : 46 Table 10: Comparison with Mixed Integer Programming Approach Voxels Beamlets #cons #vars primal dual barrier simplex simplex Brain 8,668 1,010 17,881 16,871 1,548 1,395 48 Head & Neck 14,178 1,599 27,705 26,106 31,118 4,912 58 Table 11: Run Time (seconds) for Various Linear Programming Algorithms 4.6 Solution Times One of the primary goals of our research is to automate the treatment plan generation in order to save clinicians valuable time. A related, equally important goal is to do so in a reasonably short period of time, so that more plans can be generated in a short amount of time and thus, hopefully, more patients can be treated. Therefore, our next set of experiments are aimed at assessing the run time requirements of our solution scheme. As mentioned above, we chose to develop only linear programming based technology, because linear programs can be solved efficiently. Several methodologies exist for solving linear programs, e.g., primal simplex, dual simplex, and interior point methods, and our first experiment examines the solution times for the different solution methods for our class of linear programs. We have used the linear programming solver of XPRESS (Xpress-Optimizer 2007). Table 11 presents the instance characteristics for the head and neck case and the brain case when generating a treatment plan for 8 equi-spaced beams (using a virtual critical structure) and the run times in seconds for the three linear programming algorithms (with default parameter settings). All our experiments were run on a 2.66GHz Pentium Core 2 Duo processor with 3GB of RAM under Windows XP Operating System. The differences are staggering; for this class of linear programs, interior point methods are by far superior. All remaining computational experiments therefore use XPRESS s barrier algorithm to solve linear programs. Our next set of results illustrates the impact on computation times of including a virtual critical 25