sampling issues for optimization in radiotherapy michael c. ferris r. einarsson z. jiang d. shepard
TRANSCRIPT
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Sampling Issues for Optimization in Radiotherapy
Michael C. FerrisR. Einarsson
Z. JiangD. Shepard
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Conformal Radiotherapy
• Enhanced conformation allows for greater dosages of radiation to reach the target volume (conformal shaping) while minimizing the dose delivery to surrounding normal tissues (conformal avoidance)
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Beam’s eye view
• Beam’s eye view at a given angle is determined based upon the beam source that intersects the tumor
• The view is constructed using a multi-leaf collimator
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Delivery Plan
plus some integrality constraints
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Mixed Integer Approach
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Dose/Volume Constraints
• e.g. (Langer) no more than 5% of region R can receive more than U Gy
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Alternative approaches
• Conditional Variance at Risk (CVAR)• Convex form that approximates
DVH constraints• Can use piecewise linearization and
adaptive penalty parameters• Alternatively use standard LP• P/L approach used in this work
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Wedges• A metallic wedge
filter can be attached in front of the collimator.
• It attenuates the intensity of radiation in a linear fashion from one side to other.
• Particularly useful for a curved patient surface
• 5 positions considered: Open, North, East, South, and West.
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Mixed Integer Approach
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Conformal Therapy• Conventional treatment• Beam’s eye view (collimator shaping)• Multiple angles (choose subset)• Wedges (modify intensity over field)• Non-coplanar beams (choose which planes)• Avoidance (upper bounds)• Homogeneity, conformality• Dose/volume constraints
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Assumptions/Setting• Dose calculation via Monte Carlo• Objective is “truth”; we really do
want to minimize it• Limit discussion to beam angle
selection; ideas are perfectly generalizable
• Limit “planning tool” to 3DCRT via MIP because we are nearby Europe
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Remarks
• CPLEX 9.0 used, tight tolerances• Branch/Bound/Cut code• LP relaxation solved using dual
simplex (small samples) and barrier method (large samples)
• Terma may add sparsity, CPLEX removes dense columns in factor
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Problems
• Large computational times• Large variance in computing times
• 5000-12500 sec (for 60,000 voxel case)
• Ineffective restarts (what if trials?)• Large amounts of data
• Try sampling of voxels (carefully)
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1% 3% 5% 7% 9% 11% 13% 15% 17% 19% 21%
0
200
400
600
800
1000
1200
1400
1600
Pelvis example: solution times for various sample rates;
Pro
cess
or
tim
e, [
sec]
Sampling rate
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1% 3% 5% 7% 9% 11% 13% 15% 17% 19% 21%
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
Pelvis example: objective values for various sample rates;
Ful
l o
bje
ctiv
e v
alu
e
Rate
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Naïve sampling fails• Normal tissue
• Many more voxels available• Streaking effects• Use 5x sample on 2nd largest structure
• Small structures• Minimum sample size
• Homogeneity/min/max on PTV• 2x sample on PTV, rind sampling
• Large gradients on OAR’s• 2x sample on OAR’s
• Need adaptive mechanism
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Time/quality tradeoff• Not really satisfactory• Split up problem into two phases
• Find a reduced set of angles at coarse sampling
• Optimize with reduced set of angles with finer sample
• Reduced angle problem much faster
• But doesn’t identify angle set well…
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Multiple samples
• Generate K instances at very coarse sampling rate
• Use histogram information to suggest promising angles
• How many? (e.g. K=10)• How to select promising angles?
(frequency > 20%)
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Full Objective Value• >20% scheme may lose best solution• Can calculate the objective function
with complete sample cheaply from solution of sampled problem
• Use extra information in 2 ways:1. Select only those angles that appear in the
best “full value” solutions2. Refine samples in organs where
discrepancies are greatest
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0.6 1.2 2.4 3.6 4.8 7.2 9.6 14.4 30 600
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Sample size (*1000)
Error in objective components
Err
or
totptvblarecnor
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Phase I• Must be very fast to be useful• LP relaxation much quicker
(allowing larger sample rates) and time variance much smaller
• But too many angles suggested…
• Utilize summed weight information to rank angles over complete set
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Drawbacks• Weight values not necessarily
correlated to “usefulness”• Sample objective is underestimator,
and provides little information
• Utilize this procedure to do “gross reduction”, followed by Phase II to refine angles further
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Sampling Process
• Determine initial sample size• Phase I: use all angles
• 10 sample LP’s solutions determine
• Phase II: use reduced set of angles• 10 sample MIP’s determine
• Phase III: use further reduced set • Increase sample rate, solve single MIP
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Initial sample size• Choose trial sample size K• Solve LPrelax(K)• Double K until Time(LPrelax(2*K))
unacceptable• If
unacceptable, ERROR(more time) • Value(K) is full sample objective
value from sample size K optimization
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Phase III
• Phase II may make all decisions so problem could be an LP for example
• Sample at fine enough rate to
satisfy industry requirements• Clean up phase!
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Pelvis case
• 3K prostate, 1.5K bladder, 1K rectum, 557K normal
• Time for “full problem”: 12.5K secs• Time Phase I: 16 secs• Time Phase II: 100 secs• Time Phase III: 10 secs• Solution: 40, 80, 150, 240, 270, 300
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Pancreas case
• 6K pancreas, 515 cord, 9K ltt, 6K rtt, 54K liver, 502K normal
• Time for “full problem”: 1200 secs• Time Phase I: 2 secs• Time Phase II: 12 secs• Time Phase III: 80 secs• Solution: 80, 290, 350 (+ wedges)
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Breast case
• 39K ptv, 13K heart, 11K rind breast, 71K normal
• Time Phase I: 18 secs• Time Phase II: 11 secs• Time Phase III: 2 secs• Solution: 130, 290 (+ wedges)
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Head/Neck case
• 2K ptv, 51K l/rcerebrum, 2K brainstm, 14K cerebellum, others (15-833)
• Time for “full problem”: 2542K secs• Time Phase I: 10 secs• Time Phase II: 22 secs• Time Phase III: 2 secs• Solution: 30, 140, 230 (+ wedges)
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Extensions
• Within 3DCRT• Wedges, energy levels, non-coplanar
beams all optimized concurrently
• Tomotherapy• IMAT• IMRT• Larger and more complex cases