Tarek Halabi

Approximating convex Pareto surfaces in multiobjective radiotherapy planning

Radiotherapy planning involves inherent tradeoffs: the primary mission, to treat the tumor with a high, uniform dose, is in conflict with normal tissue sparing. We seek to understand these tradeoffs on a case-to-case basis, by computing for each patient a database of Pareto optimal plans. A treatment plan is Pareto optimal if there does not exist another plan which is better in every measurable dimension. The set of all such plans is called the Pareto optimal surface. This article presents an algorithm for computing well distributed points on the (convex) Pareto optimal surface of a multiobjective programming problem. The algorithm is applied to intensity-modulated radiation therapy inverse planning problems, and results of a prostate case and a skull base case are presented, in three and four dimensions, investigating tradeoffs between tumor coverage and critical organ sparing.

Exploration of tradeoffs in intensity-modulated radiotherapy

The purpose of this study is to calculate Pareto surfaces in multi-criteria radiation treatment planning and to analyse the dependency of the Pareto surfaces on the objective functions used for the volumes of interest. We develop a linear approach that allows us to calculate truly Pareto optimal treatment plans, and we apply it to explore the tradeoff between tumour dose homogeneity and critical structure sparing. We show that for two phantom and two clinical cases, a smooth (as opposed to kinked) Pareto tradeoff curve exists. We find that in the paraspinal cases the Pareto surface is invariant to the response function used on the spinal cord: whether the mean cord dose or the maximum cord dose is used, the Pareto plan database is similar. This is not true for the lung studies, where the choice of objective function on the healthy lung tissue influences the resulting Pareto surface greatly. We conclude that in the special case when the tumour wraps around the organ at risk, e.g. prostate cases and paraspinal cases, the Pareto surface will be largely invariant to the objective function used to model the organ at risk.

Dose–volume objectives in multi-criteria optimization

Unlike conventional optimization with dose-volume (DV) constraints, multi-criteria optimization (MCO) with DV objectives provides tradeoff information which we believe is necessary for choosing better treatment plans. We show that the MCO formulation with DV objectives is better suited to convex approximation than conventional formulations with DV constraints. We provide a relaxation of the integer programming formulation which reduces the computation time for a single plan from over 5 h to about 2 min, without significantly compromising the results. We also derive a heuristic to improve on the relaxed solutions, adding only a few additional minutes of computation time. We apply these techniques to a skull based tumour case and a paraspinal tumour case. Based on a careful examination of the driving terms in the relaxed formulation and the heuristic, we argue that our techniques should apply more generally for DV objectives in multi-objective IMRT treatment planning.

Intensity modulated proton therapy for postmastectomy radiation of bilateral implant reconstructed breasts: A treatment planning study

BACKGROUND AND PURPOSE Delivery of post-mastectomy radiation (PMRT) in women with bilateral implants represents a technical challenge, particularly when attempting to cover regional lymph nodes. Intensity modulated proton therapy (IMPT) holds the potential to improve dose delivery and spare non-target tissues. The purpose of this study was to compare IMPT to three-dimensional (3D) conformal radiation following bilateral mastectomy and reconstruction. MATERIALS AND METHODS Ten IMPT, 3D conformal photon/electron (P/E), and 3D photon (wide tangent) plans were created for 5 patients with breast cancer, all of whom had bilateral breast implants. Using RTOG guidelines, a physician delineated contours for both target volumes and organs-at-risk. Plans were designed to achieve 95% coverage of all targets (chest wall, IMN, SCV, axilla) to a dose of 50.4 Gy or Gy (RBE) while maximally sparing organs-at-risk. RESULTS IMPT plans conferred similar target volume coverage with enhanced homogeneity. Both mean heart and lung doses using IMPT were significantly decreased compared to both P/E and wide tangent planning. CONCLUSIONS IMPT provides improved homogeneity to the chest wall and regional lymphatics in the post-mastectomy setting with improved sparing of surrounding normal structures for woman with reconstructed breasts. IMPT may enable women with mastectomy to undergo radiation therapy without the need for delay in breast reconstruction.

Automating checks of plan check automation

While a few physicists have designed new plan check automation solutions for their clinics, fewer, if any, managed to adapt existing solutions. As complex and varied as the systems they check, these programs must gain the full confidence of those who would run them on countless patient plans. The present automation effort, planCheck, therefore focuses on versatility and ease of implementation and verification. To demonstrate this, we apply planCheck to proton gantry, stereotactic proton gantry, stereotactic proton fixed beam (STAR), and IMRT treatments.

TU-C-T-6C-04: Quantifying the Tradeoff Between Complexity and Conformality

Purpose: Complex intensity maps in an IMRT plan may have several undesirable effects: increased leakage and integral dose, higher sensitivity to delivery errors, longer treatment time. On the other hand, the potentially negative impact of a reduction of the complexity on the dose conformality is not a priori clear. We propose to study the interplay between complexity and dose conformality in an interactive multi-objective planning approach. Method and Materials: Two surrogate measures of complexity have been studied in previous approaches: The number of multileaf collimator (MLC) segments (in step and shoot IMRT delivery), and the number of monitor units (MUs). The first is difficult to incorporate into optimization strategies because it is an “NP hard” problem and the calculation would be time prohibitive. Fortunately, one MLC vendor has proven that the MLC hardware and control software can be designed such that the pure number of MLC segments is not a limiting factor anymore. We therefore use the number of MUs as the measure of complexity. Mathematically, the number of MUs is given by the sum of positive gradients of the intensity map along the direction of leaf motion. We include the sum of positive gradients as a linear objective function in our plan optimization framework. In our approach we find solutions that are Pareto optimal, i.e., their complexity cannot be further reduced without compromising conformality. The results are incorporated into an interactive plan navigator by means of a “complexity slider”, which allows the user to interactively explore the impact of a change of the plan complexity on the dose distributions. Results: Examples of optimized IMRT plans in the head and neck region and in the prostate will be presented. The potential of reducing complexity is case dependent but in comparison with current IMRT planning approaches a substantial reduction of the number of MUs (in the order of 20%) is often achievable without compromising the dose distribution. The complexity slider allows one to explore the whole spectrum of plans including the extrema, i.e. simple but not conformal or highly conformal but complex plans. Conclusion: The complexity of an IMRT plan can be limited or reduced with various methods. In general there is a price to be paid for this. Multi-objective optimization can help to find the most suitable tradeoff. Supported by the NCI (1 R01 CA103904)

Landau and Lifshitz’ Formulation of Le Chatelier’s Principle: An Insight into Symbiosis?

A correspondence allows application of Landau and Lifshitz’ formulation of Le Chatelier’s principle from statistical physics to a simple 2-D model of biological symbiosis. The insight: symbionts stabilize the occupation of narrow peaks on fitness landscape.

SU‐FF‐T‐177: Dose‐Volume Histogram Objectives in Multi‐Criteria IMRT Optimization

Purpose: Dose‐volume histogram (DVH) constraints are frequently used in IMRT planning. For example, a DVH constraint may state that 5% (but no more) of the voxels in the planning target volume may receive a dose below the prescription level. We want to find out if the percentage of violating voxels can be reduced. We are also interested in the “price” of this reduction of violating voxels, in terms of dose to other voxels and other structures. Methods and Materials: We introduce DVH objectives into IMRT planning. Here the objective is to minimize the number of voxels that violate a given dose constraint. We then integrate DVH objectives into a multi‐criteria optimization (MCO) framework, to analyze the trade offs between DVH objectives and other planning objectives. Relaxation of mixed integer programs (MIPs) used to produce the trade off curve yields a good approximation. This is contrary to relaxation of an MIP with DVH constraints in the conventional framework. A heuristic then fine tunes the relaxation results. Results: Our methods are applied to two clinical cases with both a dose‐volume objective on the tumor and a maximum dose objective on OAR. The trade off curve between those two objectives is calculated in around 20 minutes with the relaxed MIPs compared to 40 hours with the nominal MIPs. The two techniques differ on average by only .77% tumor volume coverage and the heuristic reduces this difference to .35%. Conclusion: The use of DVH objectives (instead of DVH constraints) has the potential to lead to better trade offs in IMRTtreatment planning. Surprisingly, DVH objectives simplify the numerical handling of the problem and reduce calculation times.

SU-FF-T-116: Multicriteria IMRT Planning with Equivalent Uniform Dose (EUD) Objectives: Tumor Dose Homogeneity vs. Critical Structure Sparing

Purpose: To quantify the tradeoff between target dose homogeneity and critical structure sparing in two typical IMRT situations (prostate, para‐spinal). Furthermore, to determine the sensitivity to the response model used for critical structures (maximum vs. mean dose). Method and Materials: An EUD‐based multicriteria linear programming environment has been developed. In this work, we enforce a tumor minimum dose and compute solutions which efficiently tradeoff the tumor maximum dose and organ‐at‐risk (OAR) EUD (α⋅max dose+(1 − α) ⋅mean dose). Pareto surfaces resulting from different OAR α values are compared. The technique is applied to the RTOG horseshoe target and circular OAR geometry (varying the OARs size and location), and to two clinical cases. Results: Mathematically, if the maximum and mean doses of a structure are correlated then the choice of α does not affect the shape of the Pareto frontier. We demonstrate that this correlation is stronger for smaller OARs (a single voxel has a large impact on the mean), and also for symmetrically located OARs, which have a large set of outer ring voxels near the maximum level, as opposed to asymmetrically located OARs where the maximum dose is more localized. As the dose requirements in the tumor get more strict, we see less variance with α, since the feasible solution space is smaller. We consistently see little to no difference between Pareto surfaces for α from 0.5 to 1. Conclusion: By characterizing the conditions under which the Pareto frontier is insensitive to α, we highlight situations where it may not be necessary to know the best value of α, i.e., the exact tissue organization between purely serial and purely parallel. In general we see smooth Pareto surfaces but in some cases there were kinks pointing to outstanding treatment plans.

The double slit experiment and the time reversed fire alarm

When both slits of the double slit experiment are open, closing one paradoxically increases the detection rate at some points on the detection screen. Feynman famously warned that temptation to “understand” such a puzzling feature only draws us into blind alleys. Nevertheless, we gain insight into this feature by drawing an analogy between the double slit experiment and a time reversed fire alarm. Much as closing the slit increases probability of a future detection, ruling out fire drill scenarios, having heard the fire alarm, increases probability of a past fire (using Bayesian inference). Classically, Bayesian inference is associated with computing probabilities of past events. We therefore identify this feature of the double slit experiment with a time reversed thermodynamic arrow. We believe that much of the enigma of quantum mechanics is simply due to some variation of time’s arrow.