Sampling Issues for Optimization in Radiotherapy

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Sampling Issues for Optimization in Radiotherapy

• Michael C. Ferris
• R. Einarsson
• Z. Jiang
• D. Shepard

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Radiation Treatment Planning

• Cancer is the 2nd leading cause of death in U.S.
• Only heart disease kills more
• Expected this year in the U.S. (Amer. Cancer
  Soc.)
• New cancer cases 1.33 million (gt 3,600/day)
• Deaths from cancer 556,500 (gt 1,500/day)
• New brain/nerv. sys. cancer cases gt 18,300 (gt
  50/day)
• Cancer treatments surgery, radiation therapy,
  chemotherapy, hormones, and immunotherapy

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Radiation As Cancer Treatment

• Interferes with growth of cancerous cells
• Also damages healthy cells, but these are more
  able to recover
• Goal deliver specified dose to tumor while
  avoiding excess dose to healthy tissue and
  at-risk regions (organs)

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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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Conformal Radiotherapy
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Delivery types

• Particle beam (proton)
• Cyclotron (expensive, huge, not marketed)
• Cobalt60 based (photon)
• Gamma Knife (stereotactic, shots of radiation)
• Linear accelerator (x-ray)
• Widely available
• Conformal, IMRT, IMAT

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Linac Based Radiosurgery

• Advantages
• Cost/space
• Disadvantages
• Machine time
• Extensive QA procedures
• Reliability issues

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Example
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Notation
Dose delivered by a beam of unit weight to voxel
(i,j,k) by an angle A
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Beams eye view

• Beams 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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Dose Distribution

• Experts determine an ideal dose distribution for
  a particular target
• Covers target (tumor)
• Limits radiation to healthy/at-risk regions
• Delivery plan optimization problem

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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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Wedge effect on dose distribution
Open Beam
East Wedge
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Conformal Therapy

• Conventional treatment
• Beams 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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IMRT

• Beam collection of pencils
• Intensity maps -gt deliverable shapes -gt
  intensities of shapes
• Limit number of deliveries
• Optimize intensities and shapes concurrently
• Arc based delivery

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Commonalities

• Target (tumor)
• Regions at risk
• Maximize kill, minimize damage
• Homogeneity, conformality constraints
• Amount of data, or model complexity
• Mechanism to deliver dose

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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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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 OARs
• 2x sample on OARs
• 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 doesnt 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. K10)
• How to select promising angles? (frequency gt 20)

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Full Objective Value

• gt20 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
• Select only those angles that appear in the best
  full value solutions
• Refine samples in organs where discrepancies are
  greatest

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Three Phase Sampling

• Reduce solution time without compromising quality
• Phase I
• Sample 10 times at low rate to predict angles to
  use
• Each structure sampled proportionally with
  largest structure sample limited
• Determine angles used in best few solutions
• Phase II
• Increase sample rate, using only proposed angles
• Phase III
• Increase sample rate, fix angles and wedge
  orientations

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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 LPs solutions determine
• Phase II use reduced set of angles
• 10 sample MIPs 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(2K)) 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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Comparison (Pelvis case)

• Normal Strategy
• 41 sample, 36 angles
• 1307 secs
• Solution dvh shown
• Prostate (blue),bladder (red), rectum
  (black),normal(green)
• Three Phase Strategy
• Phase I (9 sample, 36 angles)
• 76 secs total
• Phase II (21 sample, 11 angles)
• 5 secs
• Phase III (41 sample, 6 angles)
• 3 secs
• Solutions identical
• angles 40,90,150,240,270,300

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Extensions

• Within 3DCRT
• Wedges, energy levels, non-coplanar beams all
  optimized concurrently
• Tomotherapy
• IMAT
• IMRT
• Larger and more complex cases