Principles of Robust IMRT Optimization

Timothy ChanMassachusetts Institute of Technology

Physics of Radiation Oncology – Sharpening the Edge Lecture 10

April 10, 2007

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The Main Idea

• We consider beamlet intensity/fluence map optimization in IMRT

• Uncertainty is introduced in the form of irregular breathing motion (intra-fraction)

• How do we ensure that we generate “good” plans in the face of such uncertainty?

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Outline

• “Standard optimization” vs. “Robust optimization”

• Robust IMRT

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The diet problem

• You go to a French restaurant and there are two things on the menu: frog legs and escargot. Your doctor has put you on a special diet, requiring you to get 2 units of vitamin Q and 2 units of vitamin Z with every meal.

• An order of frog legs gives 1 unit of Q and 2 units of Z

• An order of escargot gives 2 units of Q and 1 unit of Z

• Frog legs and escargot cost $10 per order.

• How much of each do you order to get the required vitamins, while minimizing the final bill? (you are cheap, but like fancy food)

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Relate back to IMRT

• Frog legs and escargot are your variables (your beamlets)

• You want to satisfy your vitamin requirements (tumor voxels get enough dose)

• Frog legs and escargot cost money (cause damage to healthy tissue)

• Objective is to minimize cost (minimize damage)

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The diet problem

• Let x = number of orders of frog legs, and y = number of orders of escargot

• The problem can be written as:

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The robust diet problem

• What if frogs and snails from different parts of the world contain different amounts of the vitamins?

• What if you get a second opinion and this new doctor disagrees with how much of vitamin Q and Z you actually need in your diet?

• How do you ensure you get enough vitamins at lowest cost?

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Robust Optimization

• Uncertainty: imprecise measurements, future info, etc.

• Want optimal solution to be feasible under all realizations of uncertain data

• Takes uncertainty into account during the optimization process

• Different from sensitivity analysis

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Lung tumor motion

• What do we do if motion is irregular?

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Towards a robust formulation

• In general, one can use a margin to combat uncertainty

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Towards a robust formulation

• In general, one can use a margin to combat uncertainty

• Uncertainty induced by motion: use a probability density function (motion pdf)

• Find a “realistic case” between the margin (worst-case) and motion pdf (best-case) concepts

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PDF from motion data

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Minimize: “Total dose delivered”

Subject to: “Tumor receives sufficient dose” “Beamlet intensities are non-negative”

Basic IMRT problem

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• To incorporate motion, convolve D matrix with a pdf…

Basic IMRT problem

Intensity of beamlet jDose to voxel i from unit intensity of beamlet j

Desired dose to voxel i

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Minimize: “Total dose delivered accounting for motion”

Subject to: “Tumor receives sufficient dose accounting for motion”

“Beamlet intensities are non-negative”

Nominal formulation

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• Introduce uncertainty in p…

Nominal formulation

Nominal pdf (frequency of time in phase k)

Dose from unit intensity of beamlet j to voxel i in phase k

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The uncertainty set

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Minimize: “Total dose delivered with nominal motion”

Subject to: “Tumor receives sufficient dose for every allowable pdf in uncertainty set”

“Beamlet intensities are non-negative”

Robust formulation

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Robust formulation

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Motivation re-visited

• Nominal problem

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Margin formulation results

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Robust formulation results

• Robust problem– Protects against uncertainty,

unlike nominal formulation

– Spares healthy tissue better than margin formulation

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Clinical Lung Case

• Tumor in left lung

• Critical structures: left lung, esophagus, spinal cord, heart

• Approx. 100,000 voxels, 1600 beamlets

• Minimize dose to healthy tissue

• Lower bound and upper bound on dose to tumor

• Simulate delivery of optimal solution with many “realized pdfs”

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Nominal DVH

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Robust DVH

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Comparison of formulations

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Numerical results

Nominal Robust Margin

Minimum dose in tumor*

94.06 %

89.25 %

99.17 %

99.87 %

100.06 %

100.07 %

Total dose to

left lung85.29 %

85.11 %

89.36 %

89.27 %

100.00 %

100.00 %

* Relative to minimum dose requirement

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Numerical results

Nominal Robust Margin

Minimum dose in tumor*

94.06 %

89.25 %

99.17 %

99.87 %

100.06 %

100.07 %

Total dose to

left lung85.29 %

85.11 %

89.36 %

89.27 %

100.00 %

100.00 %

* Relative to minimum dose requirement

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A Pareto perspective

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Continuum of Robustness

• Can prove this mathematically

• Flexible tool allowing planner to modulate his/her degree of conservatism based on the case at hand

Nominal Margin

No Uncertainty Complete Uncertainty

Robust

Some Uncertainty

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Continuum of Robustness

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Summary• Presented an uncertainty model and robust formulation to

address uncertain tumor motion

• Generalized formulations for managing motion uncertainty

• Applied the formulation to a clinical problem

• This approach does not require additional hardware