Effects of Sampling in IMRT Optimization

by Ronald L. RardinProfessor of Industrial Engineering

Purdue UniversityWest Lafayette, Indiana, USA

Caesarea Rothschild Institute, University of Haifa, June 2004

Acknowledgments• Our work at Purdue involves an inter-

disciplinary team (of 10-15) spanning– Indiana University School of Medicine– Purdue University College of Engineering– Advanced Process Combinatorics (an optimization

software firm)

• Dr. Mark Langer = our inspiration & medical mentor

• Students Felisa Preciado-Walters & Sushma Sukumaran did the computation reported here

• Sponsored in part by National Science Foundation 0120145, National Cancer Institute 1R41CA91688-01, and Indiana 21st Century Fund 830010403

External Beam Radiation Therapy

• Delivered by an accelerator that can rotate 360 degrees around the patient to treat a target at the isocenter from multiple angles

• Implemented with a Multi-Leaf Collimator varying opening during delivery time

Choices for Beamlet Intensities

Accel

erat

or

Acc

eler

ato

r

Intensity Map

(Profile)

Segments(Apertures)

Segment/Aperture Optimization• We optimize over apertures or

segments directly, with intensity maps being the weighted sum

• Find as needed via the well known column generation method of Operations Research

(Re)solve LPRelaxation of

MIP Model

Attempt toGenerate

New Apertures no more help

Round toFeasible Plan& Implement

Add ApertureColumn(s)

to Formulation

some

found

Planning Optimization Approach• Using Mixed Integer Programming (MIP),

our method explicitly enforces constraints on– Tumor dose homogeneity (ratio of min to max

tumor dose)– Maximum dose in healthy tissues– Minimum dose in secondary targets– Dose-volume limits in healthy tissues

• Then seeks segments & intensities for a fixed set of angles to maximize minimum tumor dose without violating any constraint

Model Dimensions

• Constraints – Two for each tumor point - min & max (for

homogeneity)– One for each secondary target point – lower

bound– One for each healthy tissue point - upper

bound and dose-volume (if appropriate)

• Variables– One continuous for each segment generated– One binary for each point in a dose-volume

constrained healthy tissue (rounded for solution)

Random Optimization Points• All our work has

used 1cm beamlets• We optimize over a

uniform random sample of points in each tissue– Approximate 1/4 of

the density use for fine resolution DVH’s (20-40 pts / sq cm)

– Separate on tissue boundary & interior (at least for targets)

Case ID SiteOptimPoints

EX2 Prostate 8,115

MT Prostate 9,197

FR Prostate 3,903

LUNG1 Lung 5,127

NASO Naso-Pharynx 1,863

EX3 Thorax 5,963

Post-Optimization Validation Tests

• Post-optim evaluation over a full sample generally satisfactory

• Worst on min/max or their ratio = homog, not dose-vol

Constraint Type

NumTrials

NumViols

MagnitudeOf Violation

Dose-Vol 56 3 <3% of vol

Absolute Max/Min

93 25 <3% of vol and/or

<3Gy in 24

Homog 21 5 >3% <3% of vol or <3% in

97% of vol

Speed vs. Accuracy Tradeoff

Figure D3 . Solution Time and Quality Effects of Point Density

( prostate case)

0

50

100

150

Half Base Double

MTD

(Gy)

010203040

CP

U (m

in) Computed

MTD

Actual MTD

CPU Time

Especially Need for Sensitivity

60

65

70

75

80

85

90

95

100

105

Min Tumor Dose (Gy)

6500 7000 7500 8000 8500

85%

80%

75%

70%

65%

Dose Threshold

Fraction

New Sampling Experiments

• Exploring effects of more sparse sampling in hopes of gaining solution speed without undue loss of accuracy

• Two prostate cases tested– Begin with usual base sample except double

on target (because of previous homogeneity issues)

– Optimize over combinations of 100% and 25% of these points for different classes of tissues, and boundary vs. interior

– Evaluate after optimization on the full sample

First Case: Prostate EX2

TissueBoundary

PointsInterior Points

MaxDose Dose- Volume

Target 438 4000 (maximin, homog >= 90%

Rectum 258 993 100 Gy 75% <70 Gy

Bladder 292 1000 100 Gy 80% <80 Gy

Collar 0 1566 150 Gy  

Other 0 1570 150 Gy  

Second Case: Prostate FR

TissueBoundary

PointsInterior Points

Min Dose

Max Dose

Dose- Volume

Target 515 4000 (maximim, homogeneity 85%)

Rectum   2000   100 Gy 80% <75Gy

Bladder   800   100 Gy 80% <80Gy

Femoral   1000   72 Gy 60% <50Gy

Other   916   150 Gy  

EX2 ComputationEx2 Computation

0

50

100

150

200

250

300

350

TTH

HTT

Hh

TThH

TThh

TtH

HTt

Hh

TthH

Tthh

tTH

HtT

Hh

tThH

tThh

ttHH

ttHh

tthH

tthh

Num

ber o

f Seg

men

ts

0

10

20

30

40

50

60

70

Tim

e (m

in)

ColumnsGeneratedPositiveColumns

CPU Time(Min)

FR ComputationFR Computation

0

20

40

60

80

100

120

140

160

180

TTH

TTh

TtH

Tth

tTH

tTh

ttH tth

Num

ber o

f Seg

men

ts

0

5

10

15

20

25

30

35

40

Tim

e (m

in)

Columnsgenerated

PositiveColumns

CPU Time(min)

EX2 HomogeneityEx2 Tumor

0

20

40

60

80

100

120

140

TTH

HTT

Hh

TThH

TThh

TtH

HTt

Hh

TthH

Tthh

tTH

HtT

Hh

tThH

tThh

ttHH

ttHh

tthH

tthh

Dos

e (G

y)

0

20

40

60

80

100

Per

cent

(%)

Dmin

Dmax

Homog

EX2 TumorEX2 Tumor

0

20

40

60

80

100

120

140

TTH

H

TTH

hTT

hHTT

hh

TtH

HTt

Hh

TthH

Tthh

tTH

HtT

Hh

tThH

tThh

ttHH

ttHh

tthH

tthh

Dos

e (G

y)

0

20

40

60

80

100

120

140

160

Per

cent

(%)

Dmin

D1

D5

Dmean

D95

D99

Dmax

Homog

FR HomogeneityFR Tumor

0

20

40

60

80

100

120

140

TTH

TTh

TtH

Tth

tTH

tTh

ttH tth

Dos

e (G

y)

01020304050

60708090100

Per

cent

(%)

Dmin

Dmax

Homog

FR TumorFR Tumor

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

TT

H

TT

h

TtH Tth

tTH

tTh

ttH tth

Do

se (

Gy)

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

Per

cen

t (%

)

Dmin

D1

D5

Dmean

D95

D99

Dmax

Homog

EX2 Rectum

EX2 Rectum

0

20

40

60

80

100

120

TTH

H

TTH

h

TThH

TThh

TtH

H

TtH

h

TthH

Tthh

tTH

H

tTH

h

tThH

tThh

ttHH

ttHh

tthH

tthh

Dos

e (G

y)

Dmean

D95

D99

Dmax

Ddv

FR RectumFR Rectum

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

TTH

TTh

TtH

Tth

tTH

tTh ttH tth

Dos

e (G

y)

Dmean

D95

D99

Dmax

Ddv

FR Femoral HeadsFR Femoral Heads

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

TTH

TTh

TtH

Tth

tTH

tTh ttH tth

Dos

e (G

y)

Dmean

D95

D99

Dmax

Ddv

Observations

• Small samples speed computation dramatically

• Measured tumor doses in the Target vary significantly with sampling, regardless of the exact measure used

• Small samples may be adequate in many cases to address statements about say D5 and D95, but Dmax and Dmin values will become volatile

• Dose-volume constraints are easier to satisfy with small samples that Dmax and Dmin