A new linear programming approach to radiation therapy treatment planning problems H. Edwin Romeijn∗
Ravindra K. Ahuja†
James F. Dempsey‡
December 2, 2002
Abstract We consider the problem of radiation therapy treatment planning for cancer patients. During radiation therapy, beams of radiation pass through a patient, killing both cancerous and normal cells. Thus, the radiation treatment must be carefully planned so that a clinically prescribed dose is delivered to targets containing cancerous cells, while nearby organs and tissues (called critical structures) are spared. Currently, a technique called intensity modulated radiation therapy (IMRT) is considered to be the most effective radiation therapy for many forms of cancer. In IMRT, the patient is irradiated from several beams, each of which is decomposed into hundreds of small beamlets, the intensities of which can be controlled individually. In this paper, we consider the problem of designing a treatment plan for IMRT when the orientations of the beams are given. We propose a new formulation that incorporates all aspects that control the quality of a treatment plan that have been considered to date. However, in contrast with established mixed-integer and global optimization formulations, we do so while retaining linearity of the optimization problem, and thereby ensure that the problem can be solved efficiently. Furthermore, we discuss how several more sophisticated quality and practical aspects of the problem that have been ignored to date can be incorporated into our linear model. We demonstrate the effectiveness of our approach on clinical data.
1
Introduction
Every year, approximately 1.2 million U.S. citizens are newly diagnosed with cancer (American Cancer Society [1]), and more than half of these cancer patients are treated by some form of radiation therapy (Murphy et al. [16], Perez and Brady [18]). Half of the patients treated ∗
Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, P.O. Box 116595, Gainesville, Florida 32611-6595; e-mail:
[email protected]. † Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, P.O. Box 116595, Gainesville, Florida 32611-6595; e-mail:
[email protected]. ‡ Department of Radiation Oncology, College of Medicine, University of Florida, P.O. Box 100385, Gainesville, Florida 32610-0385; e-mail:
[email protected].
1
with radiation (approximately 300,000 patients per year) may significantly benefit from conformal radiation therapy (Steel [26]). Many patients that are initially considered curable do in fact die of their disease, despite sophisticated treatment. Others may suffer unintended side effects from radiation treatment, sometimes severely reducing the quality of life. The major cause of this is that radiation treatment plans often deliver too little radiation to the targets, too much radiation to healthy organs, or both. Thus, the preservation of healthy or functional tissues, and hence the quality of a patient’s life, must be balanced against the probability of the eradication of the patient’s disease. In this paper we will propose a new linear programming based approach to designing optimal radiation treatment plans. During radiation therapy, beams of radiation pass through a patient, depositing energy along the path of the beams. This radiation kills both cancerous and normal cells. Thus, the radiation therapy treatment must be carefully planned, so that a clinically prescribed dose is delivered to cancerous cells, while sparing normal cells in nearby organs and tissues. Typically, there are several clinical targets that we wish to irradiate, and there are several nearby organs, called critical structures, that we wish to spare. Note that we usually treat targets which contain known tumors, as well as regions which contain the possibility of disease spread, or account for patient motion. If we were to treat a patient with a single beam of radiation, it might be possible to kill all the cells in the targets. However, it would also risk damaging normal cells in critical structures located along the path of the beam. To avoid this, beams are delivered from a number of different orientations spaced around the patient so that the intersection of these beams includes the targets, which thus receive the highest dose of radiation, whereas the critical structures receive radiation from some, but not all, beams and can thus be spared. Conformal radiation therapy seeks to conform the geometric shape of the delivered radiation dose as closely as possible to that of the intended targets. Recent technological advancements have lead to rapid development and widespread clinical implementation of an external-beam radiation delivery technique known as intensity modulated radiation therapy (IMRT) (see Webb [30]). IMRT allows for the creation of very complex dose distributions that allow the delivery of sufficiently high radiation doses to targets while limiting the radiation dose delivered to healthy tissues. These dose distributions are obtained by dynamically blocking different parts of the beam. The application of optimization techniques is essential to enable physicians, and thereby patients, to fully benefit from the added flexibility that this technique promises. Patients receiving radiation treatment are typically treated on a clinical radiation delivery device called a linear accelerator (see Figure 1). The radiation source is contained in the accelerator head and can be viewed as a point source of high-energy photons. The patient is immobilized with restraints on a couch, and the accelerator head can rotate for a full 360◦ about a point that is 1 meter from the source, called the isocenter. An IMRT capable accelerator is equipped with a so-called multi-leaf collimator system, which is used to generate the desired dose distribution at a given angle. Although additional degrees of freedom in movement of the couch as well as the accelerator head are available, it is common in practice to use only a single, fixed couch location, and consider a very limited number (3-9) of locations of the accelerator head in the circle around the isocenter. 2
Figure 1: Diagram of a linear accelerator. The problem of designing an optimal radiation density profile in the patient is often referred to as the fluence map optimization problem. The goal of this problem is to design a radiation treatment plan that delivers a specific level of radiation, a so-called prescription dose, to the targets, while on the other hand sparing critical structures by ensuring that the level of radiation received by these structures does not exceed some structure-specific tolerance dose. These two goals are inherently contradictory if the targets are located in the vicinity of critical structures. This is especially problematic for certain cancers, such as tumors in the head and neck area, which are often located very close to for instance the spinal cord, brainstem, and salivary glands. A common approach is to search for a radiation treatment plan that satisfies the prescription and tolerance dose requirements to the largest extent possible. All commercially available IMRT treatment planning systems, as well as most of the research in the medical physics literature to date, use local search (such as the conjugate gradient method; see, e.g., Shepard et al. [24], Xing and Chen [33], Holmes and Mackie [12], Bortfeld et al. [2]) or simulated annealing techniques (see, e.g., Rosen et al. [21], Mageras and Mohan [15], Webb [28, 29]) to find a satisfactory treatment plan. In a recent comprehensive review of the radiation therapy literature, Shepard et al. [23] surveyed many techniques previously used, including some simple linear and convex programming formulations of the problem. More recent and sophisticated operations research approaches are multi-criteria optimization (see e.g. Hamacher [10], Hamacher and K¨ ufer [11]), and mixed-integer linear programming (see e.g. Lee et al. [13, 14], Rardin [19]). Unfortunately, the trade-offs that inevitably need to be made when designing a radiation therapy treatment plan are very hard to quantify. Rather, the choice between different treatment plans is often made by a physician based on insights and experience with different types of cancers. Hamacher and K¨ ufer [11] propose a very efficiently solvable linear programming formulation of the fluence map optimization problem based on an input consisting of prescription and tolerance doses. This model is then implemented in a multi-criteria setting which interactively provides a physician with a set of candidate radiation treatment plans which can then be evaluated based on a graphical representation of the plans. In this paper we propose a new approach to the fluence map optimization problem, 3
which enriches existing models by incorporating common measures of treatment plan quality, while retaining linearity, and thereby efficient solvability, of the problem. In Section 2, we will formalize the radiation treatment planning problem and discuss various modeling issues. Then, in Section 3, we will present our formulation of the fluence map optimization problem, which balances the trade-off between radiation of the targets and critical structures, as a linear programming problem. In Section 4 we discuss the results of our approach on three clinical data sets, and compare these results with an operational planning system that is widely used. We end the paper by discussing several extensions of the model that are topics for future research in Section 5.
2 2.1
Modeling issues Notation and dose calculation
The targets and critical structures (together simply referred to as structures) are irradiated using a predetermined set of beams, each corresponding to a particular beam angle. For each beam angle, we decompose or discretize the aperture of this beam into small 1 cm2 beamlets. A 2-dimensional image is typically used to represent the position and intensity of all beamlets in a beam. Each pixel in this image is called a bixel, and its value represents the intensity (or, more correctly, fluence; see Webb and Lomax [31]) of the corresponding beamlet. We denote the set of all bixels contained in all beams by N . The core task in IMRT treatment planning is to find radiation intensities (sometimes also called weights) for all beamlets. We will denote the decision variable representing the weight of beamlet i by |N | xi , and the vector of all beamlet weights by x ∈ R+ . We denote the set of all structures by S, with targets corresponding to S 1 and critical structures to S 2 (clearly, S = S 1 ∪ S 2 and S 1 ∩ S 2 = Ø). In practice, each of the structures s ∈ S is discretized into a finite set Vs of so-called voxels. The set of all voxels will be denoted by V = ∪s∈S Vs . Note that the sets Vs are not necessarily disjoint. For instance, if a tumor has invaded a critical structure, there will be an overlap between a target and that critical structure. For convenience, we will denote the set of all voxels in target structures by V 1 = ∪s∈S 1 Vs and the set of all voxels in organs at risk and not in targets by V 2 = V \V 1 . Note that V 2 ⊆ ∪s∈S 2 Vs . (See Figure 2 for an example.) Figure 2: Targets and structures. Finally, let Pij denote the dose received by voxel j from beamlet i at unit intensity. We can then express the radiation received by each voxel as a linear function of the radiation intensities x as follows: X Dj (x) = Pij xi j ∈ V. i∈N
We will refer to these functions as the dose calculation functions.
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2.2
The Dose-Volume Histogram
The most commonly used measure used by physicians to judge the quality of a treatment plan is the so-called 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 radiation. More formally, for a given structure s ∈ S, such a histogram is a nonincreasing function Hs : R+ → [0, 1] where Hs (d) is defined as the relative volume of the part of a structure that receives d units of radiation or more. Clearly, Hs (0) = 1 and limd→∞ Hs (d) = 0. Using the finite representation of all structures using voxels, the DVH for structure s under radiation intensities x can be expressed as |{j ∈ Vs : Dj (x) ≥ d}| Hs (d; x) = . |Vs | For instance, in Figure 3 below, the point on the DVH for the target (indicated by the vertical dotted line) indicates that 90% of the target volume receives at least 70 units of radiation (usually measured in Gray (Gy)). Typically, the goal is to control or constrain the values of the DVH. In the remainder, a generic DVH will be denoted by the function H(d; x).
Figure 3: Dose-Volume Histograms.
3 3.1
Model formulation Full volume constraints
The basic constraints are hard bounds on the dose received by the structure, of the following form: (F1 ) Deliver a minimum prescription dose to all voxels in the targets (where voxels in different target volumes may have different prescription doses).
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(F2 ) Do not deliver more than a maximum tolerance dose to all voxels in both targets and critical structures (where voxels in different structures may have different tolerance doses). Constraints (F1 ) ensure a high probability of eradicating the disease, and constraints (F2 ) for the critical structures ensure that these structures are spared. In addition, constraints (ii) for the targets represent the undesirability of delivering very high doses (or very nonuniform dose distributions) to targets. Constraints of this type are often referred to as full volume constraints, since they need to be satisfied everywhere in a particular structure. The full volume constraints that need to be satisfied by a treatment plan, characterized by the beamlet intensities x, can easily be expressed as follows: X Pij xi ≥ Ls j ∈ Vs ; s ∈ S 1 i∈N
X
Pij xi ≤ Us
j ∈ Vs ; s ∈ S.
i∈N
Recall that the sets of voxels Vs are not necessarily disjoint. This means that in the current formulation, the dose of any given voxel may be subject to multiple lower and upper bound constraints. In fact, these constraints may even be conflicting. Therefore, we will assume that a ranking of the structures in order of decreasing importance exists, and will impose the hard bound constraints only for the most important structure that contains a particular voxel. In general, this will mean that any target is more important than any critical structure. The bound choice then reflects the intuition that saving a critical structure should never be at the expense of not curing the disease. For simplicity, however, we will in the remainder of this paper simply include all hard lower and upper bound constraints for all voxels.
3.2
Partial volume constraints
Unfortunately, in most practical situations, there does not exist a feasible treatment plan as defined above, as it is usually impossible to satisfy the full volume constraints for each voxel in the target volumes as well as all critical structures. There will be some voxels in the target volumes that have to be underdosed and some voxels in the critical structures that have to be overdosed. A powerful approach is to relax the full volume constraints, and then add so-called partial volume constraints that constrain the shape and location of the DVHs of all structures. Note that full volume constraints themselves can in fact be viewed as constraints on the DVHs of the structures, by realizing that (for a generic structure) they can equivalently be expressed as H(x) ≥ L H(x) ≤ U where H(x) = sup{d ∈ R+ : H(d; x) = 1} is the minimum amount of radiation that is received anywhere in the structure, and 6
H(x) = min{d ∈ R+ : H(d; x) = 0} is the maximum amount of radiation that is received anywhere in the structure. 3.2.1
Traditional formulation of partial volume constraints
In addition to the full volume constraints, we may incorporate additional constraints on the shape and location of the DVH of a structure of the following form: (P1 ) At least a fraction α of the voxels in a target volume must receive at least Lα units of radiation: H(Lα ; x) ≥ α. 0
(P2 ) At most a fraction α0 of the voxels in a structure can receive over U α units of radiation: 0
H(U α ; x) ≤ α0 . Constraints of this type can be viewed as soft bound constraints on the dose received, and are often referred to as partial volume constraints, since the corresponding bounds only need to be satisfied by a fraction of the voxels in a particular structure. To date, there have been two main approaches to deal with this issue. The first approach is to handle the nonlinearity of the constraints by reformulating them by introducing a binary variable for each voxel, which represents whether or not this voxel receives at least Lα (in the case of 0 soft lower bound constraint on a target) or at most U α units of radiation (in the case of soft upper bound constraint on a target). This approach thus results in a very large-scale integer programming problem (see e.g. Lee et al. [13, 14], Rardin [19]). The second approach is to deal with the constraints heuristically by adding an appropriate penalty term to the objective function that penalizes solutions that do not satisfy the partial volume constraints. This heuristic has been proposed by Bortfeld et al. [3], and has been incorporated into a quadratic programming-based algorithm for the fluence map optimization problem (see, e.g., Oelfke and Bortfeld [17], Wu and Mohan [32], Cho et al. [6], Spirou and Chui [25], Carol et al. [4], Deasy [8]), or a simulated annealing algorithm (see, e.g., Carol et al. [4], Cattaneo et al. [5], Cho et al. [7]). The resulting objective function is nonconvex, and the quality of the treatment plans obtained by heuristics that have incorporated this approach to handling partial volume constraints has not been established. Both approaches change the nature of the optimization problem to a nonlinear global optimization problem, or a very large-scale mixed-integer programming problem, both of which require large amounts of computation time for realistically sized problems, prohibiting a real-time solution of the models. As surrogates for the partial volume constraints, constraints that limit the variability in beamlet weights have been proposed (see Shepard et al. [23]), as well as constraints that limit the variability in the dose received by individual voxels within targets (see Hamacher and K¨ ufer [11]).
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3.2.2
A new formulation of partial volume constraints
The problem of formulating constraints on the DVH resembles to a large extent a problem that has received much attention in the financial engineering literature, in particular in risk management. Constraints on the DVH in that context are constraints on the probability distribution function of (usually investment) risks, and are notoriously difficult to solve. However, recently an alternative approach that formulates risk management constraints not in terms of the probability distribution function of risk, but rather in terms of so-called tail averages. We will apply this methodology, which has been developed in Rockafellar and Uryasev [20] and Uryasev [27], to the radiation treatment planning problem. Instead of traditional DVH constraints, we propose to incorporate, in addition to the hard lower and upper bounds on the dose received by each of the structures, alternative soft lower and upper bounds of the following form: (P01 ) The average radiation received by the subset of a target of relative volume 1 − α receiving the lowest amount of radiation must be at least equal to Lα . (P02 ) The average radiation received by the subset of a structure of relative volume 1 − α0 0 receiving the highest amount of radiation may be no more than U α . Our approach has two major advantages. From a treatment quality point of view, it does not only control the extent to which a soft lower bound is violated, but also the actual magnitude of the violation. From a computational point of view, the more straightforward approach would yield a very large-scale mixed-integer programming or global optimization problem, whereas our approach yields a linear programming problem, which can be solved much more rapidly. This would allow the user to investigate many more high quality candidate treatment plans, obtained with different soft and hard bound constraints. To formulate soft constraints of the type described above, it will be convenient to define a loss function, to allow us to treat the lower and upper bound constraints using the same methodology. For a target, low radiation doses are undesirable; we therefore define a loss function that is the negative of the dose calculation function: Lj (x) = −Dj (x)
j ∈V1
For all structures, very high levels of radiation are undesirable; the corresponding loss function is given by the dose calculation function itself: Lj (x) = Dj (x)
j ∈ V.
In the following, we will introduce and describe the concepts of Value-at-Risk and Conditional Value-at-Risk in terms of a general loss function Lj (x). The Value-at-Risk (VaR) at level α (for short, the α-VaR) is the smallest loss value with the property that no more than 100(1 − α)% of structure s experiences a larger loss. More formally, the α-VaR is defined as ζsα (x) = inf{ζ ∈ R : Hs (ζ; x) ≤ 1 − α}. 8
The Conditional Value-at-Risk (CVaR) at level α (for short, the α-CVaR) is then the average of all losses that exceed the α-VaR. More formally, and appropriately accounting for the case where Hs (ζsα (x); x) < 1 − α, the α-CVaR is defined as X 1 (Lj (x) − ζsα (x))+ . φαs (x) = ζsα (x) + (1 − α)|Vs | j∈V s
Figure 4 illustrates the concepts of VaR and CVaR for a particular dose density function. Figure 4: Value-at-Risk and Conditional Value-at-Risk. For each structure, we may now define a set of soft bounds on losses (i.e., low and high tail averages). For targets, these will be in the form of lower bounds Lαs for α ∈ As , where As is a (finite) subset of (0, 1), and upper bounds Usα for α ∈ As , where As is a (finite) subset of (0, 1) (s ∈ S 1 ). For the organs at risk, these will be in the form of only upper bounds Usα for α ∈ As , where As is a (finite) subset of (0, 1) (s ∈ S 2 ). Summarizing both the hard and soft constraints discussed so far, we are interested in finding a treatment plan x that satisfies the following constraints: j ∈ Vs ; s ∈ S 1 j ∈ Vs ; s ∈ S α ∈ As ; s ∈ S 1
Dj (x) ≥ Ls Dj (x) ≤ Us −φαs (x) ≥ Lαs α φs (x)
≤ Usα ≥ 0
xi
α ∈ As ; s ∈ S i ∈ N.
(1) (2)
where φ and φ indicates that the loss function L or L is being used, respectively. Substituting the dose calculation functions, letting zj denote the dose received by voxel j, and using the reformulation of constraints (1)-(2) as described in Rockafellar and Uryasev [20] and Uryasev [27], we obtain the following set of constraints: X Pij xi = zj j ∈ Vs ; s ∈ S i∈N
ζ αs − α
ζs +
zj ≥ Ls zj ≤ Us ´+ X³ ζ αs − zj ≥ Lαs
1 (1 − α)|Vs | j∈V s X³ 1 (1 − α)|Vs | j∈V
´ α +
zj − ζ s
≤ Usα
j ∈ Vs ; s ∈ S 1 j ∈ Vs ; s ∈ S α ∈ As ; s ∈ S 1 α ∈ As ; s ∈ S
s
xi ≥ 0 zj free α ζs free
i∈N j∈V α ∈ As ; s ∈ S 1
ζs
α ∈ As ; s ∈ S.
α
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free
3.3
Feasibility and optimality
The fluence map optimization problem has so far been formulated as a feasibility problem, i.e., it is implicitly assumed that (i) any treatment plan that satisfies all constraints is equally good; and (ii) at least one treatment plan that satisfies all constraints exists. In practice, not all plans satisfying the constraints are equally desirable. To address this issue, we will add a suitable objective function to the formulation. In particular, for each structure we will specify a penalty function of the dose received by the voxels in that structure that assigns a value to the dose received by each voxel in that structure. Denoting the penalty function for structure s by Fs we obtain the following objective function to be minimized: XX Fs (zj ). s∈S j∈Vs
Typical penalty functions that have been proposed in the literature have a value proportional to the dose received by all voxels, or only by voxels in critical structures (see Rosen et al. [22]). A more sophisticated objective function is obtained when the penalty is equal to the absolute deviation of the dose received from a prescription or tolerance dose (see Shepard et al. [23]). Since lower doses in critical structures are always preferred to higher doses, for voxels in these structures the penalty function is generally one-sided, i.e., only surpluses over the tolerance dose are penalized in the objective function. Since large deviations from the prescription or tolerance dose are considered to be much more important than small deviations, a much more frequently used alternative is to use a weighted least squares objective function. Figure 5 illustrates quadratic dose penalty functions for a target (with prescription dose 70 Gy) and a critical structure (with tolerance dose 26 Gy). To retain linearity of the optimization problem, in this paper we propose to consider piecewise linear convex penalty functions. Note that we have assumed that the penalty functions are the same for all voxels in a particular structure. In general, we may of course even relax this assumption by allowing for an individual penalty function for each voxel.
Figure 5: Quadratic dose penalty functions for a critical structure (dashed line) and a target (solid line). Unfortunately, the hard and soft bound constraints are, in practice, often conflicting. This issue can be addressed by allowing some or all of the constraints to be violated. A 10
penalization of violations of the hard lower and upper bounds on the dose received by each voxel can easily be incorporated in the penalty functions Fs , by assigning a very large penalty to surpluses or shortfalls of the dose with respect to the upper and lower bounds. In addition, we may define penalty functions for surpluses or shortfalls of the soft dose constraints, which α we will denote by Gs and Gαs , for soft upper and lower bound constraints, respectively, and for relevant combinations of a structure s and a fraction α. Summarizing, our fluence map optimization problem can be formulated as follows: XX XX α α X X Gαs (φαs ) + minimize Gs (φs ) Fs (zj ) + s∈S j∈Vs
s∈S 1 α∈As
s∈S α∈As
subject to X
Pij xi = zj
j ∈ Vs ; s ∈ S
i∈N
´+ X³ 1 α = φαs − ζ s − zj (1 − α)|Vs | j∈V s ³ ´ X 1 α α + α ζs + = φs zj − ζ s (1 − α)|Vs | j∈V
ζ αs
α ∈ As ; s ∈ S 1 α ∈ As ; s ∈ S
s
zj free xi ≥ 0 ζ αs free
j∈V i∈N α ∈ As ; s ∈ S 1
ζs
α ∈ As ; s ∈ S.
α
free
When all penalty functions are piecewise linear and convex, this problem can be formulated as a purely linear programming problem. For example, if the prescribed dose for target s ∈ S 1 is equal to ∆s ∈ [Ls , Us ], we may choose ψ(∆s − Ls ) + ψ 0 (∆s − z) if z ∈ [0, Ls ] ψ(∆s − z) if z ∈ [Ls , ∆s ] Fs (z) = ψ(z − ∆s ) if z ∈ [∆s , Us ] ψ(U − ∆ ) + ψ 0 (z − ∆ ) if z ∈ [Us , ∞) s s s 0
for appropriate parameters 0 ≤ ψ ≤ ψ 0 and 0 ≤ ψ ≤ ψ . Similarly, for appropriate critical doses ∆s ≤ Us for the critical structures s ∈ S 2 , we may choose 0 if z ∈ [0, ∆s ] ψ(z − ∆s ) if z ∈ [∆s , Us ] Fs (z) = 0 ψ(Us − ∆s ) + ψ (z − ∆s ) if z ∈ [Us , ∞) 0
again for appropriate parameters 0 ≤ ψ ≤ ψ .
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4
Results
To test our models, 3D treatment planning data for a head-and-neck cancer patient were exported from a commercial IMRT treatment planning system (Corvus, Nomos Inc.) at the University Of Florida Department Of Radiation Oncology. This data was then imported into the University Of Florida IMRT treatment planning system and used to generate the data required by the model described in Section 3. In particular, 1,232 beamlets of size 1 cm2 were generated to adequately cover the targets from five beam angles, and a voxel grid with a voxel spacing of 4mm was employed that resulted in 126,000 voxels. This generated approximately 96,000 nonzero Pij ’s in a sparse matrix of size 1,232 by 126,000 that were output by the planning system. The same five-beams (positions of the accelerator head, collimator, and couch) were used to determine a treatment plan using the commercial system. Similar constraints were used in both our model and the commercial system. Figure 6 above demonstrates both the feasibility and the potential benefits of our approach, as we were able to produce improved treatment plans while taking comparable computational times. The LP model was able to achieve a more homogeneous coverage of one of the targets (gross disease on the right of the neck) while sparing two salivary glands on the left side of the neck, whereas the commercial system was only able to spare one of the salivary glands on the left side of the neck.
Figure 6: A comparison of the treatment plans created by a commercial IMRT treatment planning system and an implementation of our model.
5
Extensions
In this section we will discuss two important issues in the determination of a good fluence map that have not been addressed to date, but can be incorporated in our model while retaining linearity of the problem.
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5.1
Fractionation
A treatment plan is never delivered to the patient in a single session, but rather as a sequence of daily treatments over an extended period of time (usually on the order of a month), to allow healthy cells to regenerate between treatments. Clinical experience has shown that a delivery of 2 Gy per session to the targets is ideal. In typical clinical cases, however, a distinction is made between two targets, usually called the Gross Tumor Volume (GTV) and the Planning Target Volume (PTV). The GTV consists of the actual identified disease, while the PTV is an expansion of the GTV to account for microscopic spread of disease as well as a margin to account for the possibility of patient and target movement, and set-up uncertainty. Clearly, the GTV is a subset of the PTV. Since the core disease is located in the GTV, this target will require a higher dose than the PTV. For instance, the GTV may require 70 Gy, while the PTV requires only 50 Gy. However, at the same time both targets should receive 2 Gy per session to ensure a high probability of eradication of the disease, which suggests a fractionation of the plan into 35 days when the GTV is taken into account, but a fractionation of the plan into 25 days when the PTV is taken into account. In the former plan the part of the PTV that is outside the GTV receives too little radiation, risking the survival of spread disease. In the latter plan, the critical structures may receive too much radiation on a single day to allow the healthy cells to regenerate between sessions, and may not be spared. The solution to this problem is to design two fluence maps, which will be delivered sequentially. The first fluence map should aim at delivering a certain prescription dose to the PTV. The second fluence map then provides additional radiation to the GTV, while avoiding the PTV. This second fluence map is often referred to as the boost. The current practice is to design the two fluence maps independently. However, this makes it hard if not impossible to control the joint effect of the treatments on, in particular, the critical structures. To ensure sparing of critical structures, it is therefore necessary to design the two fluence maps jointly. The joint design of two fluence maps that define the treatment of the patient can be incorporated into our model by doubling the decision variables for the beam intensities: variables x1i denote the beamlet intensities for the first series of treatments, while variables x2i denote the beamlet intensities for the second treatment. The set of targets K 1 is decomposed into two subsets, K 11 and K 12 , where K 11 is the set of targets (usually simply the PTV) which are the focus of the first treatment, and K 12 is the set of targets (usually simply the GTV) which are the focus of the boost treatment. We assume that K 1 = K 11 ∪ K 12 and K 11 ∩ K 12 = Ø. The model will contain full and partial volume constraints corresponding to the targets for each individual treatment, as well as full and partial volume constraints corresponding to all structures for the aggregate treatment.
5.2
Incorporating spatial effects: the 2-dimensional DVH
The DVH that is often used in practice to evaluate and compare treatment plans seems to contain a major flaw: it ignores spatial effects. In conventional conformal radiation treatment 13
planning, regions of (in particular) the target structures that receive less than some prespecified target or prescription dose are typically along the boundary of the target. Since the target usually includes some safety margin around the actual tumor, the consequences of these regions of under-radiation (often called cold spots) are usually benign. However, in IMRT treatment planning, it is very possible that such cold spots actually appear in the core of the target, which can have potentially very serious effects. Since the DVH only measures the fraction of the structure that receives at least a certain dose, it is inherently incapable of distinguishing between a treatment plan with a cold spot near the boundary of the target and a treatment plan with a cold spot near the core of the target, when both spots are similar in size and amount of radiation received. This problem is further amplified by the fact that we are dealing with 3-dimensional structures. See Dempsey et al. [9] for a discussion of this issue and for an introduction of the 2-dimensional DVH’s that we will formalize in this section. One possible avenue to try to mitigate this problem would be to assign weights to voxels, and consider a weighted DVH, using the weighted rather than actual volume. Voxels in the core of the target structure would have high weights, whereas voxels near the boundary of the target would have low weights. An alternative would be to add an extra dimension to the DVH. In particular, we propose to use a 2-dimensional DVH having both dose as well as a distance measure as arguments. The distance measure could denote the distance to the boundary of the GTV, where outside the GTV the actual distance is used, and inside the GTV the negative of the actual distance is used. The DVH would then be a function H : R+ × R → [0, 1] where H(d, r) denotes the volume of the part of a structure that receives d units of radiation and has a distance measure of at most r, relative to the volume of the part of the structure that has a distance measure of at most r. Denote the set of voxels in structure s that have a distance measure of at most r by Vs (r). Hs (d, r; x) =
|{j ∈ Vs (r) : Dj (x) ≥ d}| . |Vs (r)|
This clearly generalizes the usual concept of a DVH, since lim Hs (d, r; x) = Hs (d; x).
r→∞
Note that the 2-dimensional DVH essentially creates a DVH for each set Vs (r). It is clear that, in a similar manner as for the original DVH, we may incorporate CVaR-constraints on the 2-dimensional DVH, thereby more closely controlling the dose distribution in each structure.
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