The Challenge of Steering a Radiation Therapy Planning Optimization by Ronald L. Rardin Professor of Industrial Engineering Purdue University West Lafayette, Indiana, USA Caesarea Rothschild Institute, University of Haifa, June 2004

Acknowledgments Our work at Purdue involves an inter- disciplinary team (of 10-15) spanning –Indiana University School of Medicine –Purdue University College of Engineering –Advanced Process Combinatorics (an optimization software firm) Dr. Mark Langer = our inspiration and medical mentor Sponsored in part by National Science Foundation , National Cancer Institute 1R41CA , and Indiana 21 st Century Fund

External Beam Radiation Therapy Delivered by an accelerator that can rotate 360 degrees around the patient to treat a target at the isocenter from multiple angles Implemented with a Multi-Leaf Collimator varying opening during delivery time

Choices for Beamlet Intensities Accelerator Intensity Map (Profile)

Planning Dose Conflict Accelerator Planning seeks beamlet intensities –Sufficient dose on tumor to control it –Doses to nearby tissues within tolerances Inherent conflict between higher target dose and safety of critical healthy tissues

Optimization Limitations Optimization has made critical contributions to radiation therapy planning, but it is not a perfect fit to the dose tradeoff issues Optimization means finding a solution that minimizes (or maximizes) one function of the decision variables –Usually subject to constraints on decision choices Multiobjective optimization methods do exist, but use a sequence of single objective opts Any optimization is only practical if mathematical form of the objectives and constraints permits tractability

Interactive Sequence of Solutions Keep Limitations usually result in an interactive sequence of optimizations to find a plan suitable to physicians and dosiometrists Clinicians use graphic methods (DVH and isodose) to modify opt until they find one they like

Steerability Define steerability as the degree to which the optimization model and solution procedure are convenient for this sort of guided meta- search Purpose of this talk is to pose guidelines This interactive search is often long and frustrating –Inherently indirect as clinician changes input to an underlying optimization in order to guide it towards an acceptable plan

Typical Ingredients in Opt Models Assume the beam angles are fixed Decision variable = intensity of angle j, beamlet g Dose at point i is where are pre- computed unit dose coefficients Constraints: –Min tumor dose –Tumor homogeneity –Min 2nd target dose –Max healthy tissue dose –Dose-volume limits on healthy dose

Penalties & Importance Factors The system of constraints for given cases is almost always infeasible, i.e. there is no solution x Leads to a penalized violation format reducing all to one score to be minimized

Penalties & Importance Factors Penalty forms –Squared violation –Absolute violation –Piecewise linear violation

Optimization with a Single Score Single score fairly tractable for optimization –Unconstrained except for nonnegativity of x –Differentiable if squared penalty is used –Local minima (best only among those near) arise with dose-volume constraints but manageable Gradient methods solve quickly with squared Simulated annealing follows randomized search that adopts generated x-changes if improving & even if not with probability > 0

Steerability with a Single Score Claim the single score model is relatively poor on steerability –May input maxdose of 35 Gy in hopes of getting 45 Gy –Manipulating importance factors by hand has no guarantee of convergence –Difficult to predict what will change with requirement relaxation or tightening Will use single score to illustrate some issues

Issue 1: Meaningful Start Steered searches must start somewhere, even when constraints are inconsistent Single score model is satisfactory in this regard because constraints are enforced only through penalization in the objective function G1 Meaningful Start. Underlying optimization in a steered interactive search should guarantee a meaningful solution even if constraints are violated

Issue 2: Parameter Relevance Steering in the single score model is primarily via changing values of importance factors –This steering is indirect –Arbitrary numerical quantities without clinical import or predictable impact –Difficult and frustrating to manipulate G2 Parameter Relevance. Parameters manipulated in a steered interactive search should be meaningful to the application user

Issue 3: Objective Relevance In any optimization, relaxing (resp tightening) a requirement can only help (resp hurt) the optimal objective function value –That is, sign of impact is predictable –Local optima can confuse, but not usually too much True for single score, but impact is on the total score not on clinically relevant outcomes G3 Objective Relevance. Underlying optimization in a steered interactive search should have an objective function value meaningful to the application user

Issue 4: Hard Constraints In the single score model, all constraints are soft, i.e. weighted but not required –Hard constraints are ones explicitly enforced –Required with e.g. dose to cord <= 45 Gy Increasing may fail with squared penalty G4 Hard Constraints. The underlying optimization in a steered interactive search should be able to enforce hard constraints (or prove their inconsistency)

Issue 5: Efficient Frontier If we think of all soft constraints as objectives, we should seek a solution on the efficient frontier –No objective can be improved without hurting at least 1 other tumor dose protection of healthy tissue efficient frontier dominated solution G5 Efficient Frontier. The underlying optimization in a steered interactive search should produce a solution on the efficient frontier of soft constraints at every round

Issue 5: Efficient Frontier If 2 constraints can be simultaneously satisfied, minimizing violation may not give efficient frontier tumor dose protection of healthy tissue If 2 constraints are inconsistent an efficient solution can be obtained by minimizing violation tumor dose protection of healthy tissue

Better Paradigm: Multiobj Opt Good start: Hamacher and Kufer, “Inverse radiation therapy planning-a multiple objective optimization approach”, Discrete Applied Mathematices 118, , Considers lower bounds on target(s), upper limit on healthy tissues (but no dose-volume) –Implies opts are Linear Programs (highly tractable) Each round optimizes some weighted sum of objectives holding each to a hard limit –Optimizing instead of minimizing violation keeps solutions on efficient frontier

Better Paradigm: Multiobj Opt Paper actually proposes an automated search of the efficient frontier –Returns a collection of candidate plans Could be adapted to provide a good basis for interactive steering 1.Add some form of dose-volume constraints without conceding too much tractability 2.Use any desired starting solution procedure. Perhaps single score, or maximizing target doses for fixed healthy tissue limits (which has to be feasible) 3.At each round, select (a) criteria to focus upon (by significantly escalating their weights) and/or (b) hard bound values to tighten or relax

Better Paradigm: Multiobj Opt G1: Meaningful Start: Your choice G2: Parameter Relevance: Primarily changes in hard limits G3: Objective Relevance: Escalated weights link hard limit modification to expected changes in other criteria G4: Hard Constraints: Inherent with fixed limits G5: Efficient Frontier: Automatic with weighted sum optimization