IMPT and optimization

Editor's Notes

• #3: This chapter describes various applications of mathematical optimization techniques in treatment planning for proton therapy. The most prominent example is the optimization of beam weights for intensity-modulated proton therapy (IMPT). The conceptual and practical aspects of IMPT have been introduced in previous chapters (see primarily Chapter 11). Here, we focus on the mathematical aspects of treatment planning. First, in Section 15.1, we briefly recapitulate the basics of three-dimensional (3D) conformal therapy, including the design of spread-out Bragg peaks (SOBPs), and forward planning. In Section 15.2, we illustrate how IMPT inverse planning is formulated as a mathematical optimization problem and comment on methods to solve this problem. Then, we discuss advanced optimization techniques for proton therapy: multicriteria optimization (Section 15.3), robust optimization methods for handling range and setup uncertainty (Section 15.4), incorporation of intrafractional motion in treatment planning (Section 15.5), and consideration of radiobiological effects in IMPT optimization (Section 15.6). Finally, in Section 15.7, we review other applications of mathematical optimization in proton therapy, such as the optimization of beam current modulation and scan path for continuous scanning.
• #5: 15.1 Optimization of SOBP Fields An SOBP is the foundation of forward treatment planning for 3D-conformal proton therapy. It is used to achieve a longitudinal conformality of the required dose to the target. In his seminal paper on therapeutic use of protons, Dr. Robert Wilson recognized the need for optimization of proton dose distribution for clinical treatments, by pointing out that Bragg peaks need to spread out to uniformly cover large tumor volumes. In his assessment this could be “easily accomplished by interposing a rotating wheel of various thickness” in the beam path (1), the method of modulation that is now widely used for proton therapy (see Chapter 5). By using the word “easily,” Dr. Wilson, perhaps, anticipated the fact that the problem of optimization of modulation of SOBP would be relatively easy compared to the optimization problems yet to arise in proton therapy. 15.1.1 Optimization of Field Flatness To create a clinically relevant SOBP of the desired flatness in a passive beam scattering system, a variety of components must operate in conjunction to produce the desired beam parameters. Koehler et al. (2) described one of the earliest examples of design of flat SOBPs using computer-based optimization. Based on the input values of range and modulation width, the Treatment-Planning Optimization 463 code written in Fortran IV iteratively searched for the set of amplitudes of shifted pristine peaks, and spacings between them (in other words, relative width, and thickness of the wheel steps), which realized the desired SOBP (see Figure 15.1A). Notably, because the shape of the Bragg peak curve varies with the beam energy, the weights of individual peaks in the SOBP need to be optimized separately for different ranges in tissue, to avoid sloping in the SOBP, as shown in Figure 15.1B and C. In the early days of proton therapy, the wide variety of clinically required combinations of range and SOBP modulation required a large number of premanufactured wheels, with separate wheels required for shallow and deep tumors, one wheel for a close set of modulation width (the smallest steps of the propeller could be added or removed to allow for some variation in the total modulation width). A more flexible modern solution uses a beam current modulation system, with a limited number of wheel tracks (see Chapter 5). The pulled-back Bragg peaks can be individually controlled to produce uniform dose plateaus for a large range of treatment depths using only a small number of modulator wheels (3–5). 150 A B C D 100 50 Relative dose [%] Relative dose [%] Relative dose [%] Relative dose [%] 0 0 Depth [cm] Depth [cm] Proton SOBP Pristine peaks Proton SOBP Pristine peaks Proton SOBP Pristine peaks Proton SOBP Pristine peaks 5 10 15 20 0 5 10 15 20 0 Depth [cm] Depth [cm] 5 10 15 20 0 5 10 15 20 150 100 50 150 100 50 0 150 100 50 FIGURE 15.1 Depth–dose profile of a spread-out Bragg peak (SOBP), and constituent pristine peaks: optimization of pristine peak weights leads to (A) uniform SOBP dose, while variation in the pristine peak dose profile may introduce a (B) raising or (C) falling slope in SOBP. In principle, arbitrary profiles of the peak dose can be achieved by optimization, for example, (D) a profile with the integrated dose boost of 10% to the middle part of the SOBP. 464 Proton Therapy Physics In principle, by temporally optimizing the beam current during the modulation cycle, one can create SOBPs with arbitrary depth–dose profiles. This includes “intensity-modulated” fields according to the common definition, namely, dose distributions, that are inhomogeneous by design. Notably, the beam current modulation literally constitutes intensity modulation of the beam, regardless of whether the resulting distribution is inhomogeneous or not. An example of inhomogeneous dose achievable with range modulation is the SOBP including a simultaneous integrated dose boost delivered to a subsection of the target, as in Figure 15.1D. It should be noted though that this technique allows for intensity modulation only in depth, whereas the beam intensity is homogeneous laterally.
• #6: 15.1.2 Forward Planning with SOBP Fields Procedures of forward planning for 3D-conformal proton therapy have been well described by Bussière and Adams (6), as well as in Chapter 10 of this book. Figure 15.2 illustrates how a “manual optimization” of a treatment plan might be undertaken. The search for a satisfactory solution does involve iterative adjustment; however, it is rather subjective (e.g., depends on the planner’s training, habit, and judgment) and is not systematic (e.g., iterations do not always lead toward a more preferable solution). Thus the process cannot be termed optimization in the strictly mathematical sense. First, the irradiation directions are selected as well as the range and SOBP modulation width necessary to cover the target. Range compensators are designed to conform the dose to the distal aspect of the target, and accommodations are made to prevent underdosing of the target in case of misalignment of treatment field and tissue heterogeneities, for example, using the technique of compensator expansion, or “smearing” (7) (also see Chapter 10). Once these steps are completed, a forward calculation is performed to determine the dose from the given field, based on the assumed beam fluence. The task of the planner is then to iteratively adjust the fluences, or “weights,” of multiple beams and to combine their doses so that the resulting distribution suits a particular set of requirements. For example, in the case illustrated in Figure 15.2, irradiating the spinal cord up to the tolerance (Figure 15.2, D and G) may be considered acceptable in a certain situation, because this configuration minimizes the integral dose and the main irradiation direction is least affected by internal motion (e.g., of liver with respiration for the rightanterior beam) or variations in the stomach and bowel filling (for the left beam lateral). In other situations, such as repeat treatments, cord tolerance may be reduced, and other directions have to be used. In those cases, the clinically optimal balance, between irradiation of various structures, needs to be selected (compare, e.g., Figure 15.2, H vs. I).
• #7: 15.2 IMPT as an Optimization Problem Intensity-modulation methods allow one to achieve highest conformality of proton dose distributions to the target volume and best sparing of healthy tissue. Unlike 3D-conformal treatments, in which each SOBP field delivers a uniform (within a few percent) dose to the whole target volume, individual IMPT fields typically deliver nonuniform dose distributions (e.g., see Figure 15.3). Similar to IMRT with photons, these nonuniform field contributions combine to produce the desired therapeutic dose distribution, which may be shaped to conform to the clinical prescription. An important difference from photon intensity-modulated radiation therapy (IMRT) is that the Bragg peak of the proton depth dose distribution introduces an additional degree of freedom in modulation of the dose in depth along the beam axis, in addition to the modulation in the transverse plane, which is available in both IMRT and IMPT. Despite this difference, IMRT and IMPT are very similar regarding the mathematical formulation of the treatmentplanning problem. 466 Proton Therapy Physics To take full advantage of the possibility to sculpt the dose in depth, IMPT treatments use narrow proton pencil beams, which can be scanned across the transverse plane while changing energy and intensity to control the dose to a point. The most common and versatile IMPT technique is the 3D-modulation method, in which individually weighted Bragg peak “spots” are placed throughout the target volume (8). The examples in this chapter use 3D-modulation; however, most optimization methods described below are equally applicable to other techniques, such as single-field uniform dose (SFUD) treatments or the distal edge tracking (DET). SFUD treatments also use weighted pencil beams distributed in three dimensions, but aim at delivering a homogeneous dose to the target from every individual field direction. In the DET method, Bragg peaks are placed only at the distal surface of the tumor (9).
• #9: IMPT plan for a paraspinal tumor. CT scan showing outlines of the tumor and the spinal cord, dose distribution from a 3D IMPT treatment plan, using three beam directions. => we can observe isodose levels Dose contributions from individual beams are shown for (C) right-posterior oblique, (D) posterior, and (E) left-posterior oblique fields. (The conventional IMPT plan did not include any consideration of delivery uncertainties.) ROBUST ALGORITHMS As an illustration, consider a conventional IMPT plan for a case of paraspinal tumor, shown in Figure 15.3. The target entirely surrounds the spinal cord, which is to be spared. Total IMPT dose distribution was optimized using a quadratic objective function, thus aiming at a homogeneous target dose. As is characteristic of IMPT, the homogeneous dose distribution in the target is achieved through a superposition of highly inhomogeneous contributions delivered from three beam directions.
• #10: 5.2.1 Setup of the IMPT Optimization Problem To apply general optimization methods to radiation therapy planning, technical limitations and treatment goals need to be formulated mathematically as objectives and constraints. For that purpose, the patient image data are partitioned into volumes of interest (VOI), which could include targets, critical organs at risk of undesired side effects (organs at risk [OAR]), and other tissue volumes. VOI are further divided into basic geometric elements called voxels. The total dose distribution from an IMPT field delivered with a scanned beam can be calculated as the sum of contributions from “static” pencil beams fixed at various positions along the scan path. The dose from individual pencil beams to various voxels of interest can be represented in the form of the dose influence matrix Dij , where i is the voxel index, and j is the beam index. The total dose to any voxel is then calculated as follows: d= x D i j ij j Σ ⋅ (15.1) where xj is the relative “weight” of the beam j, which is proportional to the total number of protons delivered at the given spot, that is the position of Bragg peak. The weights xj are the optimization variables that need to be determined in treatment planning.
• #11: Because of the large number (thousands or tens of thousands) of such pencil beams involved, IMPT treatment planning requires mathematical optimization methods (10, 11). The output of the plan optimization is a set of beam weight distributions, often called intensity or fluence maps. Unlike in IMRT, where a single two-dimensional (2D) fluence map characterizes a field, in IMPT, many beam energies may be used to irradiate the target from the same direction, and optimization will yield separate maps for every energy setting.
• #12: Dosimetric or other planning objectives may be defined for volumes or individual voxels. The planning objectives and their priorities can be expressed in the objective function (OF). The term optimization, in the context of treatment planning, typically signifies the search for a set of plan parameters that minimize the value of the OF, subject to a set of constraints that have to be fulfilled. A widely used objective function that aims at minimizing the volume, within a given OAR n, that exceeds the maximum tolerance dose Dmax is given by the quadratic penalty function: Od Hd D d D n i i i OARn ( ) = ( − )( − ) ∈ Σ max max 2 (15.2) where H(d) is the Heavyside step function. Similarly, one can define a quadratic function that aims to reduce volumes of the tumor, which receive less than the minimum dose Dmin. Objective functions may also include the generalized equivalent uniform dose (12): O d N d n n i p i OARn p ( )= ( ) ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ∈ 1 Σ 1/ (15.3) 468 Proton Therapy Physics where Nn is the number of voxels in the VOI n, and p is an organ-specific parameter. In addition, there may be constraints on the dose in a VOI that have to be fulfilled in order to make the treatment plan acceptable. For example, one can request that the dose in every voxel belonging to the tumor should be between a minimum dose Dmin and a maximum dose Dmax. This would result in the hard constraint D d D i VOI i min ≤ ≤ max " ∈ . (15.4)
• #13: In clinical situations, treatment objectives often directly conflict each other: for example, a target may not be completely irradiated to the prescribed level if a dose-sensitive critical structure is immediately adjacent to it. In this case, hard dosimetric constraints have to be used with care, and it is often necessary to reformulate a constraint as an objective. For example, it may be necessary to minimize the dose to an OAR that exceeds the tolerance dose through a quadratic objective, rather than enforcing the dose to be below the maximum dose in every voxel through a constraint. Such an objective is often referred to as a “soft constraint” in the medical physics literature. Multicriteria optimization methods, discussed in Section 15.3, address such inherent treatment-planning contradictions. Thus, one can formulate the general IMPT optimization problem as follows: minimize (with respect to the beam weights x): αn n n Σ ⋅O (d) subject to the constraints: d= x D i j ij j Σ ⋅ l C d u m m m ≤ ( )≤ xj 􀁶 0. (15.5) In the above formulation the different objectives On are multiplied by respective weighting factors αn and are added together to form a single composite objective. By selecting and adjusting the weighting factors, the treatment planner can prioritize different objectives and control the trade-off between them. The approach of a weighted sum of objectives is pursued in most current treatmentplanning systems. (An alternative to this standard approach is multicriteria optimization.) The functions Cm denote a general constraint function, and lm and um are upper and lower bounds. An example of a simple constraint function is the minimum or maximum dose constraint mentioned above. Alternatively, constraints on equivalent uniform dose (EUD) can be imposed (12). Treatment-Planning Optimization 469 A number of additional parameters often need to be specified before optimization of beam weights is performed. These include, for example, the choice of the algorithm for placement of Bragg peaks (8), as well as the volume used for placement (which may be larger than the target), the spacing of peaks and layers in depth (13), and the size of the pencil beam used for delivery of therapy (14). These additional treatment parameters, or hyperparameters, affect the outcome of optimization; however, they are not determined through an optimization algorithm in the mathematical sense. Instead they are chosen based on experience, planning studies, and physical or theoretical considerations. s
• #14: 15.2.2 Solving the Optimization Problem IMPT represents a textbook example of a large-scale optimization problem, especially if convex objectives and constraints are used. The variables, the beam weights x, are continuous, that is, can take any nonnegative value (although these are often discretized, when sequenced for delivery). The objectives can typically be formulated in closed form as a function of the optimization variables, and also the gradient of the objective can be calculated analytically. Therefore a large variety of algorithms can be applied. Those can be categorized into constrained and unconstrained methods. In the case of unconstrained methods, no dosimetric hard constraints are applied, that is, all treatment goals are formulated as objectives. The only constraints that always need to be fulfilled to yield a physically meaningful plan are the variable bound constraints xj ≥ 0. However, those can be treated through relatively simple methods such as gradient projection methods. Most current treatment-planning systems use unconstrained optimization methods. In this case, improved gradient methods are used such as quasi-Newton methods or the diagonalized Newton method.
• #15: Constrained optimization for IMPT is still challenging because of the large number of variables (103 to 105) and the large number of voxels (105 to 107). If only linear objectives and constraints are used, the linear programming framework can be applied. In the nonlinear case, sequential quadratic programming methods have been used (RayStation; Raysearch Laboratories, Stockholm, Sweden) as well as barrier-penalty methods (Monaco; Elekta, Stockholm, Sweden). Optimization problems may further be classified as convex or nonconvex. In a convex optimization problem, all of the constraints as well as the minimized objective function are convex functions. For example, linear functions, and therefore, linear programming problems are convex. The feasible region (i.e., all sets of spot weights x that fulfill the constraints) is then also convex, being the intersection of convex constraint functions. With convex objectives and a convex feasible region a local optimal solution is also a global optimal solution. Thus, optimization would either yield the globally 470 Proton Therapy Physics optimal solution or demonstrate that there is no feasible solution. All the objectives and constraints described above (e.g., quadratic function, EUD) are convex. Conversely, a nonconvex optimization problem is any problem where the objective is nonconvex or nonconvex constraint functions give rise to a nonconvex feasible region. In this setting, multiple local optimal solutions are possible and, in practice, considering the large number of variables in IMPT, it is typically not possible to guarantee that an algorithm used to solve the optimization problem indeed converges to the globally optimal solution. Nonconvex constraints are becoming increasingly common in optimization. Examples of nonconvex objectives include typical radiobiological models of tumor control and normal tissue complication probabilities. Another example is the dose-volume constraints, which can be conveniently defined to specify the desired shape of the dose-volume histogram (DVH) directly (15); for example, “the fraction of the volume of a specific OAR irradiated to 40 Gy is not to exceed 30%.”
• #17: 15.3 Multicriteria Optimization Optimization theory is built up around the single criterion optimization problem, where there is one objective and other problem considerations are included as constraints. In radiotherapy, the main objective—to cover the target with the prescription dose—is in direct conflict with the other objectives of keeping the dose to the healthy organs to a minimum.
• #18: If biological response models such as tumor control probability (TCP) and normal tissue complication probability (NTCP) were reliable, one might be able to solve radiotherapy optimization well in a single criterion mode: maximize TCP subject to the NTCPs of the relevant organs at risk being below acceptable levels. However, even in this setting, depending on the patientspecific trade-off (for the treatment plan under consideration, how much gain in TCP is there if you allow NTCP for some organ to increase by some amount), were there a tool to easily explore other options, a physician might choose a different plan than the plan returned from the single-criterion optimization.
• #19: Presently, the standard commercial systems available for treatment optimization still attempt to solve the radiotherapy optimization problem with a single-criterion approach, and this leads to a lengthy optimization iteration cycle, where treatment planners try to find the set of weights and function parameters that give a plan that best matches the physician’s goals for treatment. The problem is, it is very difficult to guess those weights and function parameters to get a good plan, and as the number of organs to consider Treatment-Planning Optimization 471 increases, this task becomes increasingly more difficult. Several groups are at work to bring multicriteria optimization (MCO) into routine clinical usage (16–22). There are two main approaches to MCO for radiotherapy treatment planning: prioritized optimization and the Pareto surface (PS) approach. Below, we describe the two approaches, show how they are related, and discuss their pros and cons.
• #20: 1915.3.1 Prioritized Optimization Prioritized optimization, or lexicographic ordering, as it is sometimes called in the literature, is a natural approach for dealing with multiple objectives when the objectives can be ranked in terms of importance (23, 24). Letting O1 denote the highest priority objective, O2 the second highest, etc., prioritized optimization solves the following sequent of optimization problems for k priority levels: (1) minimize O1(x) subject to x ∈ X ; (2) minimize O2(x) subject to x ∈ X , and O1(x) ≤ O1 * · (1 + ε); … (k) minimize Ok(x) subject to x ∈ X, and, for all i < k, Oi(x) ≤ Oi * · (1 + ε), (15.6) where x is a set of the decision variables, X is a constraint set that represents constraints on the beamlet fluences (upper and lower bounds) and is also used to denote hard dosimetric constraints, such as voxel dose, organ mean dose, or EUD that must be met by every considered solution. O1 * is the optimal objective value from the first optimization (i.e., the fluence values in the case of IMPT), and ε is a small positive slip factor. Multiplication by (1 + ε) allows a small degradation in the value of the first optimization, thus hopefully permitting the second priority objective to achieve a good value, and so forth. The result of the final optimization is the single result of the prioritized optimization approach. The choice of ε (and whether it is the same for each step) and the priority ordering of the objectives will influence the final result.
• #21: 15.3.2 PS Approach The PS approach does not prioritize the objectives, but instead treats every objective equally. Unlike prioritized optimization, the PS approach yields not a single plan, but a set of optimal plans that trade off the objectives in a variety of ways. Given a set of objectives and constraints, a plan is considered Pareto-optimal if it is feasible and if there does not exist another feasible 472 Proton Therapy Physics plan that is strictly better with respect to one or more objectives and that is at least as good for the rest. Assuming that the objectives are chosen correctly, Pareto-optimal plans are the plans of interest to planners and doctors. The set of all Pareto-optimal plans comprises the PS.
• #22: The PS-based MCO problem can be formulated as follows: minimize [O1(x), O2(x), … , ON(x)] subject to x ∈ X (15.7) where X is used, as before, to represent all beamlet and dose constraints, and N is the number of objective functions. The algorithmic decisions to be made for this approach are as follows: (1) how to compute a reasonable set of diverse Pareto-optimal plans and (2) how to present the resulting information to the decision makers. Radiotherapy seems to be one of the first fields, if not the first, to fully address the question of populating PSs for N ≥ 3.
• #23: Two main types of strategies populating the PS have been put forward for the radiotherapy problem: weighted sum methods and constraint methods. Weighted sum methods are based on combining all the objectives into a weighted sum and solving the resulting scalar optimization problem. By solving the problem for a variety of weights, a variety of different Paretooptimal plans are found. If the underlying objectives and constraint set are convex, every Pareto-optimal point can be found by some weighted sum. Several publications describe methods to choose the weights appropriately, to produce a small set of plans that covers the PS sufficiently well (18, 21, 25). These methods intrinsically take into account convex combinations of calculated PS points when evaluating the goodness of a set of Pareto plans. All of these methods get bogged down when the number of objectives is large (e.g., >8). Fortunately, on a practical level, even as few as N + 1 PS plans are often sufficient to determine good treatment plans (26, 27).
• #24: Constraint methods use the objective functions as constraints (as in prioritized optimization), and by varying the constraint levels, different Pareto-optimal solutions are found. The state-of-the-art of constraint-based method is the improved normalized normal constraint (NNC) method (28). The main deficit of constraint-based methods is that error measures, which give the quality of the PS approximation, are not a natural part of the algorithm or output, as they are in the methods of Craft et al. (18) and Rennen et al. (25). Weighted sum and constraint methods are graphically depicted for 2D PSs in Figure 15.4.
• #25: 15.3.3 Navigation of the PS The final task in a PS-based approach to treatment planning is to allow the user to select a plan from the PS. Because the PS is represented by a finite set Treatment-Planning Optimization 473 of Pareto-optimal treatment plans, there are two natural approaches to plan selection. The easiest way is simply to allow the treatment planner to select one of the computed Pareto-optimal treatment plans. In the case of IMPT, where treatment plans can be weighted and combined to form other valid treatment plans, it makes sense to allow users to smoothly transition between the computed solutions. When navigating across convex combinations of the database plans, either forcing Pareto optimality or not, the standard method is to present N sliders, one for each objective, and the underlying algorithmic task is to determine how to move in the objective space in response to a slider movement (21, 29). An alternative to presenting the users with N sliders is to allow them to select two of the N objectives and then display a 2D trade-off for those two objectives. For the other N – 2 objectives, the user can impose upper bounds, influencing the 2D tradeoff surface being evaluated. The benefit of this method is that it allows the user to visualize a 2D slice of PS, which may yield intuition into the problem at hand. Figure 15.5 shows what this might look like for examining the trade-off between sparing the lung and controlling hot spots within a target. A) Weighted sum method B) e-Constraint method Normalized normal C) constraint method O2 O2 w = (.6,.4) w = (.2,.8) O2 O1 O1 O1 FIGURE 15.4 Methods to compute a database of Pareto surface points. (A) Weighted sum, (B) e-constraint, and (C) normalized normal constraint method. Target dose homogeneity OAR Sparing FIGURE 15.5 (See color insert.) Illustration of two Pareto-optimal plans, showing trade-offs in OAR sparing vs. target dose homogeneity. 474 Proton Therapy Physics
• #27: 15.3.4 Comparing Prioritized Optimization and PS-Based MCO Prioritized optimization and PS MCO are compared graphically in Figure 15.6. It is important to note that both methods rely on optimization with hard constraints. In the prioritized approach, this is obvious because objectives move into the constraint section. In the PS method, constraints are important in the problem formulation, to restrict the domain of the PS to a useful one. For example, it makes sense to put an absolute lower bound on target doses normally, even if a user is interested in exploring some underdosing of the tumor to improve OAR sparing (otherwise, anchor plans for OAR will be “all 0” dose plans, which are not helpful for planning). Similarly, a hard upper maximum dose on all voxels is useful. Therefore, MCO methods in general are best used when a constrained solver is at hand. Solvers implemented in RayStation (RaySearch Laboratories), Pinnacle (Philips Healthcare, Andover, MA), Monaco (Electa), UMPlan/UMOpt (University of Michigan, Ann Arbor, MI), and Astroid (Massachusetts General Hospital, Boston, MA) are examples of solvers that allow true hard constraints (as opposed to those that handle constraints approximately by using a penalty function with a high weight).
• #28: The advantage of the prioritized approach is that it is a programmable procedure that results in a single Pareto-optimal plan, but the disadvantage is there is only one plan presented to the user at the end of the process. PS methods on the other hand present all optimal options to the user, but might be considered overwhelming for routine planning because the user has to decide on selecting a single plan manually from the large number of options on PS. However, plan selection in standard cases may be fast, even with many options, because sliding with navigation sliders is much more efficient than the reoptimization iteration loop. Notably, because the navigation process is user-driven, it is not as reproducible as the prioritized approach.
• #30: From the delivery point of view, an optimal plan needs also to be “robust,” that is, designed in such a way that slight deviations from the plan due to various uncertainties during treatment delivery will not affect the quality of treatment outcome. In other words, a robust treatment plan will deliver a clinically acceptable dose distribution as long as the deviations from the planned do not exceed the assumed levels.
• #31: 15.4.1 IMPT Dose in the Presence of Uncertainties Doses delivered from different directions in IMPT are typically inhomogeneous and require the use of a number of proton energies. For this reason, variations in the target setup and penetration depth during delivery can lead to misalignment and mismatch of doses from individual fields, and, consequently, alter the combined dose distribution. To satisfy the requirement of dose conformity to the target, steep dose gradients are often delivered at the target border. Such steep dose gradients in the dose contributions of individual beams make IMPT plans yet more sensitive to both range and setup errors. In particular, dose gradients in the beam direction make the treatment plan vulnerable to range errors, because an error in the range of the proton beams corresponds to a relative shift of these dose contributions longitudinally inside the patient. As a consequence, the dose within the target may not add up to a homogeneous dose as desired. Hot and cold spots may arise. Moreover, dose may be shifted into critical organs. Generally, the more conformal the combined IMPT dose is, the more complex the fluence maps per field are and the more sensitive the plans are to the delivery uncertainties.
• #32: The dose distribution that results from a range overshoot of all pencil beams in this plan (i.e., protons penetrate further into the patient than anticipated during planning) would lead to a higher dose to the spinal cord, as shown in Figure 15.7A. Sensitivity of the same plan to setup errors is illustrated in Figure 15.7B, which shows the dose distribution resulting from a 3.5-mm setup error posteriorly (upwards in the picture). This shift has no impact on the dose contribution of the posterior beam. However, the oblique beams hit the patient surface at a different point. For a posterior shift, the dose contributions of the oblique beams are effectively shifted apart, which results in the cold spots around the spinal cord. From this illustration, it is evident that, unlike in conventional x-ray therapy, plan degradation in the presence of range and setup uncertainties in IMPT cannot be prevented, to a satisfying degree, with safety margins. Expanding the irradiated area around the target with margins could potentially reduce underdosage at the edge of the target in the presence of an error. However, the general problem of misaligning the dose contributions of different fields, which leads to dose uncertainties in all of the target volume, cannot be solved through margins. This problem instead relates to steep dose gradient in the dose contributions of individual fields.
• #33: 15.4.2 Robust Optimization Strategies The methods presented in this section have been described largely in three publications (32–34) that deal specifically with range and setup uncertainty in IMPT. In addition, a number of earlier publications investigate the handling of uncertainty and motion in IMRT with x-rays. Some of that work could also be applied to IMPT. For a review of developments in handling of motion and uncertainty in IMRT, see Orton et al. (35). Although this section illustrates robust optimization techniques in the context of range and setup errors, the methodology is also applicable to other types of uncertainty, for example, irregular breathing motion or uncertainty in the biological effectiveness of radiation (36). A B FIGURE 15.7 (See color insert.) Estimated dose distribution from the plan in Figure 16.3, assuming (A) a 5-mm range overshoot of all pencil beams, and (B) a systematic 3.5-mm setup error (posterior shift). Treatment-Planning Optimization 477 Several approaches that apply either the concepts of stochastic programming or robust optimization have been suggested for incorporating uncertainty into IMPT optimization. The common feature of these approaches is that the delivered dose distribution depends on a set of uncertain parameters. In the case of a rigid setup error without rotation, the set of uncertain parameters would be a 3D vector describing the patient shift in space. A simple model of range uncertainty, where it is assumed that all pencil beams simultaneously overshoot or undershoot, would have one uncertain variable that describes the range error of all beams. A more complicated model of range uncertainty could allow for different range errors for different pencil beams.
• #34: Below, we denote the set of uncertain parameters by a vector λ. The dose distribution d(x, λ) delivered to the patient depends on the beam spot weights x to be optimized, and the values of the uncertain parameters λ. The objective function used for treatment planning O(d(x, λ)) is a function of the dose distribution. 15.4.2.1 The Probabilistic Approach In the probabilistic or stochastic programming approach (34), a probability distribution P(λ), reflecting the probability for a given error to occur, is assigned to the set of uncertain parameters. Treatment plan optimization is performed by optimizing the expected value of the objective function: minimize E[O] = ∫O(d(x,λ))P(λ)dλ. (15.8) This composite objective function can be interpreted in a multicriteria view: The composite objective is a sum of objectives for every possible error scenario weighted with the probability of that error to occur. The general goal is to find a treatment plan that is good for all possible errors, but larger weights are assigned to those scenarios that are likely to occur, and lower weights to large errors that are less likely to happen. For a pure quadratic objective function, O d D i i i = Σ ( − pres )2 , the expected value of the objective function is E O = E d D +E d E d i i pres i i i [ ] [ ]− ( ) − [ ] ( ) ⎡⎣ ⎤⎦ Σ( ) 2 2 (15.9) which is the sum of two terms: the first term is the quadratic difference of the expected dose E[d] and the prescribed dose, and the second term is the variance of the dose. Hence, minimizing the expected value of the quadratic objective function aims at bringing the expected dose close to the prescribed dose in every voxel and simultaneously minimizes the variance of the dose in every voxel such that the expected dose is approximately realized even if an error occurs. 478 Proton Therapy Physics
• #36: 15.4.2.2 The Robust Approach In robust optimization (32), the values of uncertain parameters are assumed to be within some interval called the uncertainty set. Treatment planning is performed by solving the robust counterpart of the conventional IMPT optimization problem. For an introduction to robust optimization, see Ben-Tal and Nemirovski (37). Typically, this means that the constraints of the optimization problem have to be satisfied for every realization of the uncertain parameters. For example, if the original problem constrained that the maximum dose to the spinal cord be less than 50 Gray (Gy), the robust counterpart would demand that the maximum spinal cord dose is less than 50 Gy for every possible range and setup error within the uncertainty set.
• #37: For objectives, this formulation of the robust counterpart results in a worst-case optimization problem: that is, if the objective was to minimize the maximum dose to the spinal cord, then the robust counterpart would minimize the maximum spinal cord dose that can happen for any possible range or setup error. Hence, the aim is to find a treatment plan, which is as good as possible for the worst case that can occur.
• #38: 15.4.2.3 Optimization of the Worst-Case Dose Distribution Yet another approach to robust IMPT planning utilizes the concept of a worst-case dose distribution (33). This hypothetical dose distribution is defined voxel by voxel as the worst dose value that can be realized for any error anticipated in the uncertainty model. For every target voxel, the worst dose value is the minimum dose, whereas for nontarget voxels it is the maximum dose. The worst-case dose distribution is unphysical because every voxel is considered independently. Whereas in one voxel the worst case may correspond to a patient shift anteriorly, the worst case in another voxel may correspond to a patient shift posteriorly. Hence the worst-case dose distribution cannot be realized. However, it can be considered as a lower bound for the quality of a treatment plan. The method optimizes the weighted sum of the objective function evaluated for the nominal case dnom (no errors) and the objective function evaluated for the worst-case dose distribution dwc. If O is the primary objective function, then the composite objective to be optimized is given by Ocomp =O(dnom) +wO(dwc ).
• #39: 15.4.3 Examples of Robust Optimization Incorporating uncertainty in IMPT optimization yields increasingly robust treatment plans. Consider two treatment plans: a conventional plan optimized without accounting for uncertainty, and a plan optimized for range and setup uncertainty using the probabilistic approach (i.e., the setup and range uncertainties modeled with a Gaussian distribution). Figure 15.8 shows the DVHs corresponding to dose distributions calculated for range Treatment-Planning Optimization 479 and setup errors randomly sampled from these Gaussian distributions. For the conventional plan, target coverage is strongly degraded in many cases, and the dose to the spinal cord can be very high for some scenarios. The variation in the DVHs of the robust plan is greatly reduced, ensuring better target coverage and lower spinal cord doses. To gain some insight into how this robustness is achieved, let us consider the dose contributions of individual beams. Figure 15.9 compares four treatment plans: the conventional plan, a plan optimized for range uncertainty only, a plan optimized for setup uncertainty only, and a plan incorporating both types of errors. The conventional plan is characterized by steep dose 100 80 60 40 20 00 10 20 30 40 Conventional Robust 50 60 70 80 90 FIGURE 15.8 DVH comparison between a conventional and a robust IMPT plan. DVHs for the CTV and the spinal cord are shown for randomly sampled range and setup errors. A B C D FIGURE 15.9
• #40: For the case illustrated in Figure 15.3, dose contributions from the posterior beam from four differently optimized plans. (A) Conventional IMPT, robust IMPT incorporating (B) range uncertainty only, (C) setup uncertainty only, and (D) considering both range and setup uncertainty. 480 Proton Therapy Physics gradients both in beam direction and laterally, especially around the spinal cord. The plan optimized for range uncertainty shows reduced dose gradients in beam direction and avoids placing a steep distal falloff of a Bragg peak in front of the spinal cord. The lateral falloff is used instead of the distal falloff to shape the dose distribution around the spinal cord. The plan optimized for setup errors only shows reduced dose gradients in the lateral direction, but it does not avoid placing a distal Bragg peak falloff in front of the critical structure and therefore does not provide robustness against range errors per se. The plan optimized for both range and setup errors shows reduced dose gradients both longitudinally, in the beam direction, and laterally. In summary, robustness is achieved through a redistribution of dose contributions among the beam directions and through avoiding unfavorable dose gradients. For our sample paraspinal case, the price of robustness is a higher dose to the spinal cord for the nominal case. In a conventional plan, the steep distal Bragg peak falloff is utilized, which allows for optimal sparing of the spinal cord. If range errors are to be accounted for, the shallower lateral falloff is used, leading to a more shallow dose gradient between tumor and spinal cord for the nominal case. Publications by Pflugfelder et al. (33) and Unkelbach et al. (34) provide a more detailed analysis. In the experience of the authors, all of the methods to account for uncertainty, described above, lead to similar treatment plans and may be equally suited to account for systematic uncertainties.
• #42: 15.5 Temporospatial (4D) Optimization Precision of therapy delivery can be affected not only by the changes in setup and patient anatomy between treatment fractions, but also by the intrafractional motion of the target, which could be due to respiration, peristalsis, or organ settling due to gravity (see Chapter 14 for more details). If no action is taken, there is always a risk that parts of the target may move outside of the treatment field, resulting in a loss of dose coverage. Even in cases where treatment-planning margins are generous enough to cover the full amplitude of motion, intrafractional motion would degrade dose gradients and increase irradiation of surrounding healthy tissues. An important difference from x-ray therapy is that, in particle therapy, because of the limited range, the use of margin expansions, such as internal target volumes, requires explicit consideration of possible changes in radiological depth to target, because these are often affected by organ motion (38). Additionally, as with x-rays, in dynamically delivered intensity-modulated therapy, certain patterns of superposition of motion of the target and the scanned beam, or so-called “motion interplay,” can have a severe impact on the delivered dose (e.g., 39). Treatment-Planning Optimization 481
• #43: Numerous ideas have been put forward that aim to mitigate the impact of intrafractional motion: these include recommendations for selection of planning image set, compensator expansion, internal margins (40, 41), delivery methods using beam gating (42), field rescanning (43), and target tracking (44). In this section, we review approaches to incorporate intrafractional motion into the optimization of beam weights in IMPT. Those methods have been investigated primarily in the context of IMRT with photons. Although the methodology can be transferred to IMPT, those approaches have not been validated in detail regarding the specific challenges mentioned above, that is, interplay effects and sensitivity to changes in radiological path length. Methods to incorporate intrafractional motion in plan optimization require a characterization of the geometrical variation of the patient’s anatomy. For respiratory motion, this can be obtained from respiratory-correlated computed tomography (CT) (often called 4D CT), which provides the geometry of the patient in several phases of the breathing cycle (45). The task of evaluating the actual dose distribution delivered to a moving target requires first calculating instantaneous dose to all phases of the 4D CT. Figure 15.10 illustrates variation in the proton dose distribution delivered to a changing anatomy, throughout the respiratory cycle. Such instantaneous doses can then be mapped onto a reference anatomical set, by using the correspondence established between the voxels of different CT sets, obtained through elastic image registration. The mapped dose can be subsequently added along with contributions from all instances of variable anatomy, to yield the dose accumulated throughout the respiratory cycle (46, 47). Instantaneous dose on phase-specific CT Full inhalation Full A B C D E exhalation Mid-ventilation Instantaneous dose mapped to exhalation CT FIGURE 15.10 Dosimetric evaluation of a treatment plan for a tumor in the liver, using respiratory-correlated CT. Estimated instantaneous dose delivered during (A) the full exhalation, (B) full inhalation, and (C) mid-ventilation phases of the respiratory cycle. To estimate the total dose, contributions from various instances of the anatomy have to be mapped onto the reference CT set, for example, for (D) full inhalation dose (dose “B” mapped onto the full exhalation CT “A”), and (E) mid-ventilation (dose “C” mapped onto CT “A”). (CT images courtesy of Dr. S. Mori (NIRS). With permission.)
• #44: 15.5.1 Plan Optimization Based on a Known Motion Probability Density Function In a simple approach to include motion, it is assumed that motion is sufficiently well described by the reconstructed phases of a 4D CT and that the dose delivered to a voxel i is obtained by summation of the dose contributions from all phases: d= p d = p x D i r r i r r r j ij r j Σ ( ) ( ) Σ ( )Σ ⋅ ( ) . (15.10) Here, r is an index to the instance of geometry, and the voxel index i refers to an anatomical voxel defined in the reference phase. Dij (r) is the dose influence matrix for phase r. Its calculation requires elastic registration of the CT of phase r with the reference phase. The parameters p(r) are probabilities that the patient is in phase r and are referred to as the motion probability density function (PDF), which can be estimated from a recorded breathing signal. Treatment planning can be performed by optimizing the beam weights xj based on objectives and constraints evaluated with the cumulative dose from Equation 15.10 above (48). The general idea is that, rather than passively letting the motion deteriorate the original plan, one should anticipate it, and, in fact, actively engage it in shaping the desired dose distribution. The resulting treatment plan would deliver an inhomogeneous dose distribution to a static geometry. However, the inhomogeneities are designed such that, after accumulating dose over the whole breathing cycle, the desired dose distribution is obtained. Because one of the most manifest effects of motion on the dose is the smoothing or washout of gradients both within the target and at its borders, the logical way to counter this effect is dose boosting at the edges of the target, in what is termed “edge-enhancement” (49). The exact pattern of optimum inhomogeneity enhancement is determined by the form of motion PDF. Generally, the effect of motion on the dose may be approximated as convolution of the dose with the PDF; thus, the desired motion-compensated plan can be roughly approximated with the inverse process: deconvolution. However, this is constrained by the requirement that the fluences delivered at all pencil beam spots are physical; thus, if negative values arise from deconvolution or during optimization, those need to be reset to zero (or the minimum should be allowed, if the beam cannot be completely turned off, e.g., in a continuous scan). Because the PDF does not depend on time, the use of probabilistic planning does not require complex technical delivery modifications to ensure synchronization of the beam with the motion cycle, and thus delivery of such fields can be relatively easily implemented in practice. However, PDF-based optimization methods rely on the reproducibility of target motion patterns during delivery, and sufficient sampling of the motion PDF. When motion deviates from the expectation, a significant dosimetric deviation may occur.
• #46: Treatment planning for proton therapy usually uses a constant relative biological effectiveness (RBE) factor of 1.1 for the conversion of physical dose di to “biological” dose (see Chapter 19). The biological effective dose is defined as the photon dose from a 60Co source that would produce the same cell-kill in the tumor. Under the assumption of a constant RBE, treatment plan optimization can be performed based on the physical dose alone as described in the preceding sections of this chapter. In other words, the physical dose is the only measure that is needed to characterize the radiation field and to assess the quality of the treatment plan. However, this may be an oversimplification and a second quantity may be needed to characterize the radiation field and its radiobiological effectiveness. This second measure is the linear energy transfer (LET) (see Chapters 2 and 19). Radiobiological experiments suggest that the amount of radiation-induced cell-kill increases with higher LET, and consequently at the end of range of the proton beams. To directly incorporate effects of varying RBE in IMPT planning, the objective function needs to be formulated in terms of physical dose and LET, instead of dose alone. One approach has been suggested by Wilkens and Oelfke (52), who formulate their objective function based on the linear-quadratic cell survival model, where the α-parameter depends 484 Proton Therapy Physics linearly on LET. Recently, Grassberger et al. (53) have demonstrated that it is feasible in IMPT optimization to influence the distribution of LET without significantly altering the physical dose distribution.
• #48: 15.7.1 Scan Path Optimization In 3D spot scanning, beam spots are typically placed on a regular grid over the tumor region. In practice though, a large number of beam spots will be assigned zero weight in the optimization of the treatment plan. Nevertheless, in a naïve implementation of spot scanning, the beam would be steered in a zigzag pattern over the entire grid, including the spot positions that correspond to zero weight. Kang et al. (54) investigated the optimization of the scan path of the beam in order to avoid regions with zero weight spots. The problem corresponds to a “traveling salesman” problem and simulated annealing has been applied to solve the problem.
• #49: 15.7.2 Beam Current Optimization for Continuous Scanning There are different ways to perform pencil beam scanning. In spot scanning, the proton beam is steered to one desired position on the grid, delivers dose according to the optimized spot weight, is switched off, and is moved to the next grid point. In continuous scanning, the beam is constantly moving according to a predefined pattern. The intensity-modulated field is delivered by modulating the beam current in time while the beam is repeatedly scanned over the tumor volume. In this case, an additional computational step is needed that converts the optimized spot weights defined at discrete positions to the beam-current modulation that approximately delivers the same fluence. For this step, optimization methods have been applied (14); however, this optimization can be performed in fluence space. It does not require dose calculation in the patient and is therefore easier to solve than the optimization of spot weights.