Principles of Robust IMRT Optimization

1. Principles of Robust IMRT Optimization Timothy Chan Massachusetts Institute of Technology Physics of Radiation Oncology – Sharpening the Edge Lecture 10 April 10, 2007

2. The Main Idea • We consider beamlet intensity/fluence map optimization in IMRT • Uncertainty is introduced in the form of irregular breathing motion (intra-fraction) • How do we ensure that we generate “good” plans in the face of such uncertainty?

3. Outline • “Standard optimization” vs. “Robust optimization” • Robust IMRT

4. The diet problem • You go to a French restaurant and there are two things on the menu: frog legs and escargot. Your doctor has put you on a special diet, requiring you to get 2 units of vitamin Q and 2 units of vitamin Z with every meal. • An order of frog legs gives 1 unit of Q and 2 units of Z • An order of escargot gives 2 units of Q and 1 unit of Z • Frog legs and escargot cost $10 per order. • How much of each do you order to get the required vitamins, while minimizing the final bill? (you are cheap, but like fancy food)

5. Relate back to IMRT • Frog legs and escargot are your variables (your beamlets) • You want to satisfy your vitamin requirements (tumor voxels get enough dose) • Frog legs and escargot cost money (cause damage to healthy tissue) • Objective is to minimize cost (minimize damage)

6. The diet problem • Let x = number of orders of frog legs, and y = number of orders of escargot • The problem can be written as:

7. The robust diet problem • What if frogs and snails from different parts of the world contain different amounts of the vitamins? • What if you get a second opinion and this new doctor disagrees with how much of vitamin Q and Z you actually need in your diet? • How do you ensure you get enough vitamins at lowest cost?

8. Robust Optimization • Uncertainty: imprecise measurements, future info, etc. • Want optimal solution to be feasible under all realizations of uncertain data • Takes uncertainty into account during the optimization process • Different from sensitivity analysis

9. Lung tumor motion • What do we do if motion is irregular?

10. Towards a robust formulation • In general, one can use a margin to combat uncertainty

11. Towards a robust formulation • In general, one can use a margin to combat uncertainty • Uncertainty induced by motion: use a probability density function (motion pdf) • Find a “realistic case” between the margin (worst-case) and motion pdf (best-case) concepts

12. PDF from motion data

13. Basic IMRT problem Minimize: “Total dose delivered” Subject to: “Tumor receives sufficient dose” “Beamlet intensities are non-negative”

14. Basic IMRT problem • To incorporate motion, convolve D matrix with a pdf… Dose to voxel i from unit intensity of beamlet j Intensity of beamlet j Desired dose to voxel i

15. Nominal formulation Minimize: “Total dose delivered accounting for motion” Subject to: “Tumor receives sufficient dose accounting for motion” “Beamlet intensities are non-negative”

16. Nominal formulation • Introduce uncertainty in p… Dose from unit intensity of beamlet j to voxel i in phase k Nominal pdf (frequency of time in phase k)

17. The uncertainty set

18. Robust formulation Minimize: “Total dose delivered with nominal motion” Subject to: “Tumor receives sufficient dose for every allowable pdf in uncertainty set” “Beamlet intensities are non-negative”

19. Robust formulation

20. Motivation re-visited • Nominal problem

21. Margin formulation results

22. Robust formulation results • Robust problem • Protects against uncertainty, unlike nominal formulation • Spares healthy tissue better than margin formulation

23. Clinical Lung Case • Tumor in left lung • Critical structures: left lung, esophagus, spinal cord, heart • Approx. 100,000 voxels, 1600 beamlets • Minimize dose to healthy tissue • Lower bound and upper bound on dose to tumor • Simulate delivery of optimal solution with many “realized pdfs”

24. Nominal DVH

25. Robust DVH

26. Comparison of formulations

27. Numerical results * Relative to minimum dose requirement

28. Numerical results * Relative to minimum dose requirement

29. A Pareto perspective

30. Continuum of Robustness • Can prove this mathematically • Flexible tool allowing planner to modulate his/her degree of conservatism based on the case at hand Nominal Robust Margin No Uncertainty Some Uncertainty Complete Uncertainty

31. Continuum of Robustness

32. Summary • Presentedan uncertainty model and robust formulation to address uncertain tumor motion • Generalized formulations for managing motion uncertainty • Applied the formulation to a clinical problem • This approach does not require additional hardware