1. Radiotherapy Planning Stephen C. Billups University of Colorado at Denver http://www-math.ucdenver.edu/~billups sbillups@carbon.ucdenver.edu

2. Goals Deliver enough radiation to a tumor to destroy the tumor. Minimize damage to the patient. Bad News:Radiation must travel through healthy tissue to get to the tumor.

3. Radiation Delivery

4. Guiding Principles Healthy tissue can recover from small doses. –So, hit the tumor from different directions. Avoid hitting critical organs.

6. A Treatment Plan

7. Radiotherapy Planning Determine which gantry angles to use For each angle used, determine –How much radiation to deliver –How to “shape” the radiation beam

8. Shaping the Radiation Beam Multileaf Collimator

9. Outline Physics of radiation oncology The geometry of Radiotherapy Dose deposition operator A “Simple” linear programming model for radiotherapy planning Other issues: –Dose-volume constraints –Minimum-support plans –Dynamic planning –Uncertainty issues. Summary/Conclusions

10. Some physics High energy photons, through collisions, set fast electrons in motion, which Kick atomic electrons off molecules, which Lead to chemical reactions, which Lead to impaired biological function of DNA, which Leads to cell death

11. Fluence number of crossing photons Fluence = --------------------------------- surface area crossed

12. Fluence vs. dose Fluence is exponential in depth Dose is nearly exponential Dose Fluence

13. Geometry

14. Terminology Beam – A cone emanating from the accelerator and enclosing the entire target area. (Corresponds to a single gantry position). Pencil – Part of a beam, along which a nearly constant dose is delivered. Pixel/Voxel – Smallest subdivision of the target area. Pixel=square, Voxel=cube

15. Dose Deposition Operator As a pencil of radiation passes through the body, it deposits a certain fraction of its energy in each pixel it passes through. The dose deposition operator specifies what fraction of each pencil is deposited in each pixel.

16. Dose Deposition Operator For pixel i, beam b, pencil p

17. Dose Deposition Operator where x(p,b) is the intensity of pencil p in beam b.

18. Dose Deposition Operator The dose deposition operator allows for accurate modeling of the physics. –nonlinearities due to depth of penetration –scattering –etc. But the resulting optimization model is still linear! (Tractable) –so long as dose is proportional to beam intensity

19. Linear Programming Model not linear

20. Linear Programming Model Standard Trick

21. More Simply

22. Other Goals Dose Volume Constraint: –No more than x % of a structure can exceed “y” dose. Minimum support plans. –Keep the number of gantry angles small Dynamic plans. Uncertainty

23. Dose Volume Histogram

24. Dose Volume Constraint M is a really big number N is the maximum number of pixels in the organ that can get “fried”. Integer Constraint=Hard

25. Dose Volume Constraint The Integer Programming formulation is too hard for general purpose solvers to solve –Requires specialized code. –Don’t try this at home!

26. Minimum Support Plans See: S.C. Billups and J. M. Kennedy, Minimum-Support Support Solutions for Radiotherapy Planning, Annals of Operations Research (to appear).

27. Many beams used Expensive to administer

28. Few beams. Clinically, the plan is nearly as good. Practical to administer.

29. Finding Minimum-support Plans

30. Integer Programming Formulation

31. Exponential Approximation Approximate *-norm by exponential function.

32. Exponential Approximation

33. Successive Linearization Algorithm 1.Solve LP model 2.Linearize exponential problem around latest solution (generating new LP) 3.Solve new model. 4.Repeat steps 2 and 3 until solution stops changing.

34. Penalized Subproblems If a beam is “barely” turned on in one solution, it will be penalized heavily in the next subproblem.

35. Choosing Parameters β balances the therapeutic goals against the number of beams used. α controls the size of beams that are penalized significantly. –Large α – only weakest beams are penalized –Small α – all beams penalized to varying degrees. –If α is large enough, exponential problem has same solution as integer programming problem.

36. Quality of Solutions Successive linearization algorithm generates a local solution to the exponential problem. Compared to integer programming solution, the SLA solution may use slightly more beams, but is just as good clinically.

37. Uncertainty The dose actually delivered differs from the plan: –Modeling approximations –Patients move during treatment! –etc. How sensitive are solutions to this uncertainty? Can we devise more robust models?

38. Dynamic Planning Radiation is delivered over 20 days. (Same drill every day). Is it possible to measure the effects of the plan and adjust the plan each day? See Ferris and Voelker. Neuro-dynamic programming for radiation treatment planning, Numerical Analysis Research Report NA-02/06, Oxford University Computing Laboratory, Oxford University, 2002.

39. Summary/Conclusions Good IP algorithms exist for doing 3- dimensional planning with difficult constraints. Handling uncertainty was the biggest concern at the workshop last February Also, growing interest in dynamic planning.

40. More Information http://www.trinity.edu/aholder/HealthApp/oncology