Transcript Robust IMRT Optimization

Principles of
Robust IMRT Optimization
Timothy Chan
Massachusetts Institute of Technology
Physics of Radiation Oncology – Sharpening the Edge
Lecture 10
April 10, 2007
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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Basic IMRT problem
Minimize: “Total dose delivered”
Subject to: “Tumor receives sufficient dose”
“Beamlet intensities are non-negative”
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Basic IMRT problem
Dose to voxel i from unit
intensity of beamlet j
Intensity of beamlet j
Desired dose
to voxel i
• To incorporate motion, convolve D matrix with a pdf…
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Nominal formulation
Minimize: “Total dose delivered accounting for motion”
Subject to: “Tumor receives sufficient dose
accounting for motion”
“Beamlet intensities are non-negative”
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Nominal formulation
Dose from unit intensity of
beamlet j to voxel i in phase k
Nominal pdf (frequency of
time in phase k)
• Introduce uncertainty in p…
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The uncertainty set
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Robust formulation
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”
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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
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•
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•
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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 lung
85.29 %
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 lung
85.29 %
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
Nominal
No Uncertainty
Robust
Some Uncertainty
Margin
Complete Uncertainty
• Can prove this mathematically
• Flexible tool allowing planner to modulate his/her degree of
conservatism based on the case at hand
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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
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