Laura Goadrich13 Dec 2004 A Metaheuristic for IMRT Intensity Map Segmentation Laura D. Goadrich October 15, 2004 Supported with NSF Grant DMI-0400294

Laura Goadrich13 Dec 2004

A Metaheuristic for IMRT Intensity Map Segmentation

Laura D. GoadrichOctober 15, 2004

Supported with NSF Grant DMI-0400294

Contents• Motivation• Radiotherapy: Conformal vs. IMRT• Intensity Map & Shape Matrices• Program Outline

– Constraints– Difference Matrix

• Results• Improving Solvability

– Partitioning – Condor– Nested Partitions

• Future Works• References






Radiation treatment of cancer: a bit of trivia….

• Radiation has been used to treat cancer for more than 100 years. In fact, the first cancer patient was treated in Chicago in January, 1896, less than one month after the discovery of X-rays.

• Intensity modulated radiation therapy (IMRT) is a revolutionary type of external beam treatment that is able to conform radiation to the size, shape and location of a tumor.

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)).

Goals of Radiotherapy

1. Apply radiation to tumor (target volume) sufficient to destroy it while maintaining the functionality of the surrounding organs (organs at risk)

2. Minimize amount of time patient spends positioned and fixed on the treatment couch.

3. Minimize beam-on time (time in which radiation is applied to patient)

Planning Radiotherapy- Tumor Volume Contouring

QuickTime™ and aNone decompressor

are needed to see this picture.

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).

Planning Radiotherapy- Beam Angles and Creating Intensity Maps



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








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

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).



multileaf collimator






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.

0 0 80 100 100 80 40 00 80 100 80 60 100 100 400 80 60 60 60 80 40 400 100 60 60 60 60 100 6060 60 80 80 80 80 80 020 40 20 20 40 80 20 00 100 60 80 100 100 100 00 40 80 100 80 80 0 00 0 60 100 40 0 0 0

Angle 55˚

Delivery of an Intensity Map via Shape Matrices

0 40 60 60 40 0 040 60 40 40 20 40 040 40 40 40 40 40 4040 40 40 40 40 40 4040 40 40 20 40 40 020 40 20 40 40 60 00 60 40 40 40 0 0

0 1 1 1 1 0 00 1 1 1 1 1 01 1 1 1 1 1 11 1 0 0 0 0 00 1 1 1 1 1 00 0 0 0 0 1 00 0 0 0 0 0 0

0 1 1 1 0 0 01 1 0 0 0 0 01 1 0 0 0 0 01 0 0 0 0 0 01 0 0 0 0 0 01 1 1 1 1 1 00 1 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 1 00 0 0 1 1 1 10 0 1 1 1 1 11 1 1 0 0 0 00 1 0 0 0 0 00 1 1 1 1 0 0

0 0 1 1 1 0 01 1 1 1 0 0 00 0 1 0 0 0 00 1 1 1 1 1 10 0 0 0 1 1 00 0 0 1 1 1 00 1 1 1 1 0 0

Original Intensity Map

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

+++

x 20 x 20 x 20x 20

=






Program Input/Output

• Input: – An mxn intensity matrix A=(ai,j) comprised of

nonnegative integers

• Output: – T aperture shape matrices dt (with entries dt

ij)

– Non-negative integers t (t=I..T) giving corresponding beam-on times for the apertures

– Apertures obey the delivery constraints of the MLC and the weight-shape pairs satisfy

€

tdt = A

t=1

T

∑

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

€

min α t = Zt=1

T

∑

€

min gt = Gt=1

Z

∑

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

numShapeMatrices*7 + beam-on time

• Initializing T close to the optimal number of matrices + 1 required reduces the solution space and solution time






Intensity Map as Sum of Shapes

I = k Sk

K

k=1

k > 0 is time the linear accelerator is opened to release uniform radiation

Sk is shape matrix

Intensity Matrix = Sum of Shapes (Sk) times their weights (k)

Multileaf Collimator (MLC) problem with minimal beam-on time

min t

subject to t St = I t

where t is an element ofthe index set of all possibleshape matrices

t

t

Multileaf Collimator (MLC) problem with minimal beam-on time

min t + (K - 1)Tc

subject to t St = I

t

where t is an element ofthe index set of all possibleshape matricesTc is set-up timeK is the number of shapes used

t

t

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

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.

€

pijt + lij

t =1− dijt

pijt , lij

t ,dijt ∈ {0,1}

ptij= 1 if beam element in

row i, column j is covered by the right leaf when the tth monitor unit is delivered = 0 otherwiselt

ij is similar for the right leafdt

ij =1 if bixel is open

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).

€

pijt ≤ pij +1

t

lij +1t ≤ lij

t0 1 0 1 0 0

NON-CONTIGUOUS

shape matrix:

leaf setting:0 1 1 1 0 0

CONTIGUOUS

shape matrix:

leaf setting:

Constraint: No Leaf Collisions

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

0 0 0 1 0 00 1 0 0 0 0

0 0 0 1 0 00 0 1 0 0 0

COLLISION

NO COLLISION

shape matrix:

leaf setting:

shape matrix:

leaf setting:

€

li+1, jt + pij

t ≤1

li−1, jt + pij

t ≤1

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

€

dijt

j=1

J

∑i=1

I

∑ ≤ z t × I × J

z ∈ {0,1}

Must sum to the intensity matrix.

is the intensity assigned to

beam element dt

ij

€

tdijt

t=1

T

∑ = Aij

€

t

Constraint: MonoshapeNo rows gaps are allowed: monoshapes are required• First determine which rows in each monitor unit are open to

deliver radiation

€

deliveryit ≤ dijt ≤ delivery it

j=1

Ncols

∑

delivery ∈ {0,1}

deliveryit=1 if the ith row is being used a time t = 0 otherwise

Determine if the preceding row in the monitor unit delivers radiation

€

deliveryi−1,t − delivery it ≤ dropit

drop ∈ {0,1}

dropit=1 if the preceding row (i-1) in a shape is non-zero and the current row (i) is 0 = 0 otherwise

Constraint: Monoshape

• Determine when the monoshape ends

€

deliveryit − delivery i−1,t ≤ jumpit

jump∈ {0,1}

jumpit=1 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

€

jumpit ≤1i= 2

Nrows

∑

€

dropit ≤1i= 2

Nrows

∑

€

deliveryi+1,t ≤1− dropIt

I = 2

Nrows

∑

Complexity of Problem

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

type level Lowest Num Consts Avg Num Consts Largest Num Constsprostate 5 2178 2707 3267prostate 10 3889 4838 5841

head&neck 5 3257 3519 3695head&neck 10 5511 6231 6606head&neck 100 56555 64800 72012pancreas 5 5518 6432 6687pancreas 10 9112 10961 13839






Diff: Heuristic

• Fast heuristics use a difference matrix

• Transformation: Given an mxn intensity matrix M, define the corresponding mx(n+1) 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, j+1) - M(i, j)

Difference Matrix example

0 0 80 100 100 80 40 00 80 100 80 60 100 100 400 80 60 60 60 80 40 400 100 60 60 60 60 100 6060 60 80 80 80 80 80 020 40 20 20 40 80 20 00 100 60 80 100 100 100 00 40 80 100 80 80 0 00 0 60 100 40 0 0 0

Angle 55˚

0 0 80 20 0 -20 -40 -40 00 80 20 -20 -20 40 0 -60 -400 80 -20 0 0 20 -40 0 -400 100 -40 0 0 0 40 -40 -6060 0 20 0 0 0 0 -80 020 20 -20 0 20 40 -60 -20 00 100 -40 20 20 0 0 -100 00 40 40 20 -20 0 -80 0 00 0 60 40 -60 -40 0 0 0

Difference Matrix example

20 40 2040 80 6060 60 800 40 60

20 20 -20 -2040 40 -20 -6060 0 20 -800 40 20 -60

Diff in Practice

• Variables:– Delta: generates difference matrix– Count: counts nonzero rows– Frequency(D,v): counts appearances of v or -v in matrix D

• AlgorithmD = delta(M) // generate initial difference matrixwhile (count(D) > 0){

find d > 0 that maximizes frequency(D,d) // choose intensity dcall create_shape_matrix(S,d) // create shape matrix S

D= D - d*delta(S) // update the difference matrix}






Comparison of Results: Prostate Case for Corvus 4.0

Weighted Score = numShapeMatricies*7 + beam-on time

Weighted Scores for Level 5 Intensity Maps

0

100

200

300

400

500

600

35 80 135 225 280 325

Angles

Weighted Score

Corv4

Dif3

BC30

BC120


0

100

200

300

400

500

600

700

35 80 135 225 280 325

Angles

Weighted Score

Corv4

Dif3

BC30

BC120

DNR DNR DNR


0

100

200

300

400

500

600

700

35 80 135 225 280 325

Angles

Weighted Score

Corv4

Dif3

BC30

BC120

DNR DNR DNRDNR DNRDNR

Comparison of Results: Head & Neck Case for Corvus 4.0

Weighted Score for Level 5 Intensity Maps

0

100

200

300

400

500

600

700

55 165 245 290 350

Angles

Weighted Score

Corv4

Dif3

BC30

BC120

DNR DNR


0

200

400

600

800

55 165 245 290 350

Angles

Weighted Score

Corv4

Dif3

BC30

BC120

DNRDNR


0100200300400500600700800

55 165 245 290 350

Angles

Weighted Score

Corv4

Dif3

BC30

BC120

DNR DNRDNR DNR DNR

Comparison of Results: Pancreas Case for Corvus 4.0

Weighted Score Level 5 Intensity Maps

0

200

400

600

800

1000

0 51 103 154 206 257 308

Angles

Weighted Score

Corv4

Dif3

BC30

BC120

DNR DNR DNR DNR DNR DNR DNR


0

200

400

600

800

1000

0 51 103 154 206 257 308

Angles

Weighted Score

Corv4

Dif3

BC30

BC120



0

200

400

600

800

1000

1200

0 51 103 154 206 257 308

Angles

Weighted Score

Corv4

Dif3

BC30

BC120







Delivery of an Intensity Map via Shape Matrices

0 40 60 60 40 0 040 60 40 40 20 40 040 40 40 40 40 40 4040 40 40 40 40 40 4040 40 40 20 40 40 020 40 20 40 40 60 00 60 40 40 40 0 0

0 1 1 1 1 0 00 1 1 1 1 1 01 1 1 1 1 1 11 1 0 0 0 0 00 1 1 1 1 1 00 0 0 0 0 1 00 0 0 0 0 0 0

0 1 1 1 0 0 01 1 0 0 0 0 01 1 0 0 0 0 01 0 0 0 0 0 01 0 0 0 0 0 01 1 1 1 1 1 00 1 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 1 00 0 0 1 1 1 10 0 1 1 1 1 11 1 1 0 0 0 00 1 0 0 0 0 00 1 1 1 1 0 0

0 0 1 1 1 0 01 1 1 1 0 0 00 0 1 0 0 0 00 1 1 1 1 1 10 0 0 0 1 1 00 0 0 1 1 1 00 1 1 1 1 0 0

Original Intensity Map

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

+++

x 20 x 20 x 20x 20

=

Improving computation time via divide-and-conquer

0 1 1 1 1 0 00 1 1 1 1 1 01 1 1 1 1 1 11 1 0 0 0 0 00 1 1 1 1 1 00 0 0 0 0 1 00 0 0 0 0 0 0

0 1 1 1 0 0 01 1 0 0 0 0 01 1 0 0 0 0 01 0 0 0 0 0 01 0 0 0 0 0 01 1 1 1 1 1 00 1 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 1 00 0 0 1 1 1 10 0 1 1 1 1 11 1 1 0 0 0 00 1 0 0 0 0 00 1 1 1 1 0 0

0 0 1 1 1 0 01 1 1 1 0 0 00 0 1 0 0 0 00 1 1 1 1 1 10 0 0 0 1 1 00 0 0 1 1 1 00 1 1 1 1 0 0

partition and match upper and lower shapes

+++

x 20 x 20 x 20x 20

0 1 1 1 0 0 01 1 0 0 0 0 01 1 0 0 0 0 01 0 0 0 0 0 0

0 1 1 1 1 0 00 1 1 1 1 1 01 1 1 1 1 1 11 1 0 0 0 0 0

0 1 1 1 1 1 00 0 0 0 0 1 00 0 0 0 0 0 0

1 0 0 0 0 0 01 1 1 1 1 1 00 1 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 1 00 0 0 1 1 1 10 0 1 1 1 1 1

1 1 1 0 0 0 00 1 0 0 0 0 00 1 1 1 1 0 0

0 0 1 1 1 0 01 1 1 1 0 0 00 0 1 0 0 0 00 1 1 1 1 1 1

0 0 0 0 1 1 00 0 0 1 1 1 00 1 1 1 1 0 0

Recreate full shapes by matching upper shapes to lower shapes

0 1 1 1 1 0 00 1 1 1 1 1 01 1 1 1 1 1 11 1 0 0 0 0 00 1 1 1 1 1 00 0 0 0 0 1 00 0 0 0 0 0 0

0 1 1 1 0 0 01 1 0 0 0 0 01 1 0 0 0 0 01 0 0 0 0 0 01 0 0 0 0 0 01 1 1 1 1 1 00 1 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 1 00 0 0 1 1 1 10 0 1 1 1 1 11 1 1 0 0 0 00 1 0 0 0 0 00 1 1 1 1 0 0

0 0 1 1 1 0 01 1 1 1 0 0 00 0 1 0 0 0 00 1 1 1 1 1 10 0 0 0 1 1 00 0 0 1 1 1 00 1 1 1 1 0 0

partition and match upper and lower shapes

+++

x 20 x 20 x 20x 20

0 1 1 1 0 0 01 1 0 0 0 0 01 1 0 0 0 0 01 0 0 0 0 0 0

0 1 1 1 1 0 00 1 1 1 1 1 01 1 1 1 1 1 11 1 0 0 0 0 0

0 1 1 1 1 1 00 0 0 0 0 1 00 0 0 0 0 0 0

1 0 0 0 0 0 01 1 1 1 1 1 00 1 0 0 0 0 0

0 0 0 0 0 0 00 0 0 0 0 1 00 0 0 1 1 1 10 0 1 1 1 1 1

1 1 1 0 0 0 00 1 0 0 0 0 00 1 1 1 1 0 0

0 0 1 1 1 0 01 1 1 1 0 0 00 0 1 0 0 0 00 1 1 1 1 1 1

0 0 0 0 1 1 00 0 0 1 1 1 00 1 1 1 1 0 0






Condor: Increasing Throughput• Created by UW-Madison CS department

– Software and documentation is Free– Supports Unix, Linux, Windows

• Workload management system for compute-intensive jobs

• Runs on clusters- using idle computers• Provides:

– Job queuing mechanism– Scheduling policy– Priority scheme– Resource monitoring – Resource management

• Allows serial or parallel jobs

Condor: www.cs.wisc.edu/condor

• Only need a Submission file and a Code file (with any input files- stdin & file input)– Sample Submission file

Condor: timely response

• Sample execution of 5 programs submitted simultaneously to Condor






Nested Partitions

1. Partitioning- create a neighborhood- Partition subspace by identifying shapes

2. Random Sampling- Create random shapes- Use the random shape with a given probability

3. Promising Region- All solutions using the chosen shape- Valued based on Price of the best full solution

4. Backtrack- Disallow a shape to be used

• A shape is created by choosing– Each row and a value– Growing each row upward and downward

• Use all possible columns in each new row• Eg. The optimal result is using 2 shapes

• Red is the starting cell.

NP: random shapes

0 20 00 40 2040 40 0

=0 20 00 20 2020 20 0

0 0 00 20 020 20 0

+

= + +0 0 00 40 040 40 0

0 20 00 0 00 0 0

0 0 00 0 200 0 0

• Selecting each cell individually would result in the worst case scenario of having 8 different shapes

• A good random shape can be created by – Choosing a random (non-zero) starting row– Choosing a random starting and ending column

(could be the same column) without holes– Growing the row up and down (storing each new

shape).

NP: random shape

0 20 00 40 2040 40 0

NP: storing shapes• Heap benefits

– Quick, easy retrieval– Quick sorting of price (best price at top of tree)– Can get a flat sample (eg. every 10th shape)

• Heap disadvantages– No easy way to select a biased sample– Gives no feel for the amount of different types of

shapes (size, variety of prices)

• Therefore, for selecting random shape, need a Bucket sorter

NP: bucket sorter

• Benefit: have information on each bucket within easy access – Amount of shapes,type,etc.

• Heap would have to keep a record of the types and amounts of types entered– Looses the heap benefits of speed – Too much overhead

NP: bucket sorter

• Can get a biased sample– Uses the knowledge that shapes with better prices

turn out to create good solutions.

• Linear

• Exponential: weighted distribution

n(n+1)2

Amount from each bucket

=

60% from best price30% from next best10% from next best

NP: regions

Initial SolutionS1

1, S21, S3

1, …

Promising Region (1)Force S1

1 New Solution: S1

2, S22, S3

2, …

Complementary Region (1)Disallow S1

1

New Solution: S1

3, S23, S3

3, …

Shapes organized best (price) to worst by increasing subscript.

… … … …PR1 > CR1 PR1 <= CR1

NP: promising region


1 New Solution: S1

2, S22, S3

2, …

Promising Region(2) Force S1

2, S22

New Solution: S1

4, S24, S3

4, …

Complementary Region(2) Allow one of S1

2 or S22

New Solution: S1

5, S25, S3

5, …

… … … …PR2 <= CR2PR2 > CR2

NP: complementary region


3, S23

New Solution: S1

6, S26, S3

6, …

Complementary Region (3) Allow one of S1

3 or S23

New Solution: S1

7, S27, S3

7, …

Complementary Region (1)Disallow S1

1

New Solution: S1

3, S23, S3

3, …

… … … …PR3 > CR3 PR3 <= CR3

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.

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.

Documents

Laura Goadrich13 Dec 2004 A Metaheuristic for IMRT Intensity Map Segmentation Laura D. Goadrich October 15, 2004 Supported with NSF Grant DMI-0400294