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

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

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

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

17 The uncertainty set

18 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” Robust formulation

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 NominalRobustMargin Minimum dose in tumor * % % % % % % Total dose to left lung % % % % % * Relative to minimum dose requirement

28 Numerical results NominalRobustMargin Minimum dose in tumor * % % % % % % Total dose to left lung % % % % % * 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 NominalMargin No UncertaintyComplete Uncertainty Robust Some Uncertainty

31 Continuum of Robustness

32 Summary Presented an 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