The Challenge of Steering a Radiation Therapy Planning Optimization

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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

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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
  0120145, National Cancer Institute
  1R41CA91688-01, and Indiana 21st Century Fund
  830010403

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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

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Choices for Beamlet Intensities
Intensity Map (Profile)
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Planning Dose Conflict

• 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

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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

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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

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Steerability

• 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

• 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

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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

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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

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Penalties Importance Factors

• Penalty forms
• Squared violation
• Absolute violation
• Piecewise linear violation

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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 gt 0

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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

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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
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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
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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
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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 lt 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)
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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
efficient frontier
dominated solution
protection of healthy tissue
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
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Issue 5 Efficient Frontier

• If 2 constraints can be simultaneously satisfied,
  minimizing violation may not give efficient
  frontier

• If 2 constraints are inconsistent an efficient
  solution can be obtained by minimizing violation

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Better Paradigm Multiobj Opt

• Good start Hamacher and Kufer, Inverse
  radiation therapy planning-a multiple objective
  optimization approach, Discrete Applied
  Mathematices 118, 145-161, 2002.
• 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

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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
• Add some form of dose-volume constraints without
  conceding too much tractability
• Use any desired starting solution procedure.
  Perhaps single score, or maximizing target doses
  for fixed healthy tissue limits (which has to be
  feasible)
• At each round, select (a) criteria to focus upon
  (by significantly escalating their weights)
  and/or (b) hard bound values to tighten or relax

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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