1. Effects of Sampling in IMRT Optimization by Ronald L. Rardin Professor of Industrial Engineering Purdue University West Lafayette, Indiana, USA Caesarea Rothschild Institute, University of Haifa, June 2004

2. 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 21 st Century Fund 830010403

3. 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

4. Choices for Beamlet Intensities Accelerator Intensity Map (Profile)

5. 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 LP Relaxation of MIP Model Attempt to Generate New Apertures no more help Round to Feasible Plan & Implement Add Aperture Column(s) to Formulation some found

6. 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

7. 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)

8. 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 IDSite Optim Points EX2Prostate 8,115 MTProstate 9,197 FRProstate 3,903 LUNG1Lung 5,127 NASONaso-Pharynx 1,863 EX3Thorax 5,963

9. 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 Num Trials Num Viols Magnitude Of Violation Dose-Vol563<3% of vol Absolute Max/Min 9325<3% of vol and/or <3Gy in 24 Homog215 >3%<3% of vol or <3% in 97% of vol

10. Speed vs. Accuracy Tradeoff

11. Especially Need for Sensitivity

12. 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

13. First Case: Prostate EX2 Tissue Boundary Points Interior Points Max DoseDose- Volume Target4384000(maximin, homog >= 90% Rectum258993100 Gy75% <70 Gy Bladder2921000100 Gy80% <80 Gy Collar01566150 Gy Other01570150 Gy

14. Second Case: Prostate FR Tissue Boundary Points Interior Points Min Dose Max Dose Dose- Volume Target5154000(maximim, homogeneity 85%) Rectum 2000 100 Gy80% <75Gy Bladder 800 100 Gy80% <80Gy Femoral 1000 72 Gy60% <50Gy Other 916 150 Gy

15. EX2 Computation

16. FR Computation

17. EX2 Homogeneity

18. EX2 Tumor

19. FR Homogeneity

20. FR Tumor

21. EX2 Rectum

22. FR Rectum

23. FR Femoral Heads

24. 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