A Metaheuristic for IMRT Intensity Map Segmentation

1
A Metaheuristic for IMRT Intensity Map
Segmentation

• Athula Gunawardena, Warren DSouza,
• Laura D. Goadrich, Kelly Sorenson, Robert Meyer,
  and Leyuan Shi
• University of Wisconsin-Madison
• October 15, 2004
• Supported with NSF Grant DMI-0400294

2
Radiotherapy Motivation

• 1.2 million new cases of cancer each year in
  U.S., and many times that number in other
  countries
• Approximately 40 of U.S. patients with cancer
  have radiation therapy sometime during the course
  of their disease
• Organ and function preservation are important
  aims (minimize radiation to nearby organs at risk
  (OAR)).

3
Planning Radiotherapy- Tumor Volume Contouring

• Isolating the tumor from the surrounding OAR
  using CAT scans is vital to ensure the patient
  receives minimal damage from the radiotherapy.
• Identifying the dimensions of the tumor is vital
  to creating the intensity maps (identifying where
  to focus the radiation).

4
Planning Radiotherapy- Beam Angles and Creating
Intensity Maps

• Multiple angles are used to create a full
  treatment plan to treat one tumor.

5
Option 1 Conformal Radiotherapy

• The beam of radiation used in treatment is a 10
  cm square.
• Utilizes a uniform beam of radiation
• ensures the target is adequately covered
• however difficult to avoid critical structures
  except via usage of blocks

6
Option 2 IMRT

• Intensity Modulated Radiotherapy (IMRT) provides
  an aperture of 3mm beamlets using a Multi-Leaf
  Collimator (MLC), which is a specialized,
  computer-controlled device with many tungsten
  fingers, or leaves, inside the linear
  accelerator.
• Allows a finer shaped distribution of the dose to
  avoid unsustainable damage to the surrounding
  structures (OARs)
• Implemented via a Multi-Leaf Collimator (MLC)
  creating a time-varying aperture (leaves can be
  vertical or horizontal).

7
IMRT Planning- Intensity Map

• There is an intensity map for each angle
• 0 means no radiation
• 100 means maximum dosage of radiation
• Multiple beam angles spread a healthy dose
• A collection of apertures (shape matrices) are
  created to deliver each intensity map.

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Delivery of an Intensity Map via Shape Matrices
Original Intensity Map

Shape Matrix 1
Shape Matrix 3
Shape Matrix 2
Shape Matrix 4



x 20
x 20
x 20
x 20
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Program Input/Output

• Input
• An mxn intensity matrix A(ai,j) comprised of
  nonnegative integers
• Output
• T aperture shape matrices dt (with entries dtij)
• Non-negative integers ?t (tI..T) giving
  corresponding beam-on times for the apertures
• Apertures obey the delivery constraints of the
  MLC and the weight-shape pairs satisfy

10
Mechanical Constraints

• After receiving the intensity maps, machine
  specific shape matrices must be created for
  treatment.
• There are numerous types of IMRT machines
  currently in clinical use, with slightly
  different physical constraints that determine the
  possible leaf positions (hence the possible shape
  matrices).
• Each machine has varying aperture setup times
  that can dominate the radiation delivery time.
• To limit patient discomfort and patient motion
  error reduce the time the patient is on the
  couch.
• Goals
• Minimize beam-on time
• Minimize number of different shapes

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Approach Langer, et. al.

• Mixed integer program (MIP) with Branch and Bound
  by Langer, et. al. (AMPL solver)
• MIP linear program with all linear constraints
  using binary variables
• Langer suggests a two-phase method where
• First minimize beam-on time
• T is an upper bound on the
• number of required shape matrices
• Second minimize the number of segments (subject
  to a minimum beam-on time constraint)
• gt 1 if aperture changes
• 0 otherwise

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In Practice

• Langer, et. al. do not report times and we have
  found that computing times are impractical for
  many real applications.
• To obtain a balance between the need for a small
  number of shape matrices and a low beam-on time
  we seek to minimize
• numShapeMatrices7 beam-on time
• Initializing T close to the optimal number of
  matrices 1 required reduces the solution space
  and solution time

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Constraint Right and Left Leaves Cannot Overlap

• To satisfy the requirement that leaves of a row
  cannot override each other implies that one beam
  element cannot be covered by the left and right
  leaf at the same time.

ptij 1 if beam element in row i,
column j is covered by the right leaf
when the tth monitor unit
is delivered 0 otherwise ltij is similar
for the right leaf dtij 1 if bixel is open
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Constraint Full Leaves and Intensity Matrix
Requirements

• Every element between the leaf end and the side
  of the collimator is also covered (no holes in
  leaves).

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Constraint No Leaf Collisions

• Due to mechanical requirements, in adjacent rows,
  the right and left leaves cannot overlap

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Accounting and Matching Constraints

• The total number of shape matrices used is
  tallied.
• zt 1 when at least one beam
  element is exposed
• when the tth monitor unit in
• the sequence is delivered
• 0 otherwise
• I is the number of rows
• J is the number of columns

• Must sum to the intensity matrix.
• is the intensity assigned to
• beam element dtij

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Constraint Monoshape

• No rows gaps are allowed monoshapes are
  required
• First determine which rows in each monitor unit
  are open to deliver radiation

deliveryit1 if the ith row is being used
a time t 0 otherwise

• Determine if the preceding row in the monitor
  unit delivers radiation

dropit1 if the preceding row (i-1)
in a shape is non-zero and the
current row (i) is 0 0 otherwise
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Constraint Monoshape

• Determine when the monoshape ends

jumpit1 if the preceding row (i-1)
in a shape is zero and the current
row (i) is nonzero 0 otherwise

• There can be only one row where the monoshape
  begins and one row to end

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Complexity of Problem

• The complexity of the constraints results in a
  large number of variables and constraints.

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Diff Heuristic

• Fast heuristics use a difference matrix
• Transformation Given an mxn intensity matrix M,
  define the corresponding mx(n1) difference
  matrix D
• Expand M by adding a column of zeros to the left
  and to the right sides of M
• Define D row-wise by the differences D(i, j)
  M(i, j1) - M(i, j)

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Diff in Practice

• Variables
• Delta generates difference matrix
• Count counts nonzero rows
• Frequency(D,v) counts appearances of v or -v in
  matrix D
• Algorithm
• D delta(M) // generate initial difference
  matrix
• while (count(D) 0)
• find d 0 that maximizes frequency(D,d) //
  choose intensity d
• call create_shape_matrix(S,d) // create shape
  matrix S
• D D - ddelta(S) // update the difference
  matrix

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Comparison of Results Prostate Case for Corvus
4.0
Weighted Score numShapeMatricies7
beam-on time
23
Comparison of Results Head Neck Case for
Corvus 4.0
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Comparison of Results Pancreas Case for Corvus
4.0
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Future Work

• Incorporate the Nested Partitions method into our
  shape matrix method to take advantage of
  randomized strategies.
• Partition the more complicated shapes into two
  smaller shapes which can be handled quickly and
  easily. Then merge the resulting segments using
  the marriage algorithm to give a solution to the
  original problem.

26
Referenced Papers

• N. Boland, H. W. Hamacher, and F. Lenzen.
  Minimizing beam-on time in cancer radiation
  treatment using multileaf collimators. Networks,
  2002.
• T.R. Bortfeld, D.L. Kahler, T.J Waldron and
  A.L.Boyer, X-ray field compensation with
  multileaf collimators. International Journal of
  Radiation Oncology Biology 28 (1994), pp.
  723-730.
• T. Bortfeld, et. al. Current IMRT optimization
  algorithms principles, potential and
  limitations. Massachusetts General Hospital,
  Harvard Medical School, Presentation 2000.
• D. Dink, S.Orcun, M. P. Langer, J. F. Pekny, G.
  V. Reklaitis, R. L. Rardin, Importance of
  sensitivity analysis in intensity modulated
  radiation therapy (IMRT). EuroInforms
  Presentation 2003.
• K. Engel, A new algorithm for optimal multileaf
  collimator field segmentation. University
  Rostock, Germany, March 2003.
• M. Langer, V. Thai, and L. Papiez, Improved leaf
  sequencing reduces segments or monitor units
  needed to deliver IMRT using multileaf
  collimators. Medical Physics, 28(12), 2001.
• P. Xia, L. J. Verhey, Multileaf collimator leaf
  sequencing algorithm for intensity modulated
  beams with multiple static segments. Medical
  Physics, 25 (8), 1998.