A Metaheuristic for IMRT Intensity Map Segmentation Athula Gunawardena, Warren D’Souza, Laura D....

A Metaheuristic for IMRT Intensity Map Segmentation

Athula Gunawardena, Warren D’Souza,

Laura D. Goadrich, Kelly Sorenson, Robert Meyer, and Leyuan Shi

University of Wisconsin-Madison

October 15, 2004

Supported with NSF Grant DMI-0400294

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)).

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).



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 55Þ

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

t1

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

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 t Zt1

T

min gt Gt1

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

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 pij1

t

lij1t 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:

li1, 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

j1

J

i1

I

z t I J

z {0,1}

Must sum to the intensity matrix.

is the intensity assigned to

beam element dtij

tdijt

t1

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

delivery it dijt delivery it

j1

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 deliveryit 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 1i2

Nrows

dropit 1i2

Nrows

deliveryi1,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)

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

Wei

gh

ted

Sco

re Corv4

Dif3

BC30

BC120


0

100

200

300

400

500

600

700

35 80 135 225 280 325

Angles

Wei

gh

ted

Sco

re Corv4

Dif3

BC30

BC120

DNR DNR DNR


0100

200300400500

600700

35 80 135 225 280 325

Angles

Weig

hte

d S

core

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

Wei

gh

ted

Sco

re Corv4

Dif3

BC30

BC120

DNR DNR


0

200

400

600

800

55 165 245 290 350

Angles

Weig

hte

d S

core Corv4

Dif3

BC30

BC120

DNRDNR


0100200300400500600700800

55 165 245 290 350

Angles

Weig

hte

d S

core 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

Weig

hte

d S

core 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

Weig

hte

d S

core Corv4

Dif3

BC30

BC120



0

200

400

600

800

1000

1200

0 51 103 154 206 257 308

Angles

Weig

hte

d S

core Corv4

Dif3

BC30

BC120


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.

A Metaheuristic for IMRT Intensity Map Segmentation Athula Gunawardena, Warren D’Souza, Laura D....

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