1. Medical considerations to guide mixed integer formulations of IMRT planning problems from MD to NP to CR Mark Langer, MD Radiation Oncology Indiana University Medical Center

2. Formulations and fabulations j=1 a i=1,j=1 j=2 a i=1,j=2 d i=1 Bahr, GK Radiol, 1968 Sonderman, Abrahamson, Op Res 1985; Lee Langer, M, Int J Rad Onc 1990

3. Problem Scale Beams – 9 to 36 –may need to choose “best” 9 of 36 Beamlets per beam –100 – 400 Structures –3-6 Constraint tolerances –3% at any point, or within 3mm if dose gradient ≥ 30%/cm Optimization bound tolerance –1 fraction size (about 2.5%)

4. Voxel dose calculated on a grid.25mm X 0.25mm on a slice, slices 0.5 cm apart 10 cm wide.25 cm Dose distribution – matrix stored Dose distribution - isodose plot Dose distribution – dose volume histogram 10 cm long 1600 grid pts/slice X 20 = 32 000 4 -6 critical structures Treatment volume

5. What kind of accuracy is expected? What is the Point to Take home? Dose at individual points – recommended agreement calculations vs measured: –Low dose gradient (<30%/cm) region: 4% dose –High dose gradient region: 4mm Dyk, J. Van et al, for the Comm. Of Medical Physicists of the Ontario Cancer Institute and Foundation Int. J. Radiat. Oncol. Biol. Phys. 26:261-73(1993)

6. What kind of accuracy is expected? Dose at individual points Measured vs. calculated doses –“The phantoms will be designed to assess the accuracy of a delivered dose (+5%) near the center of the target” Cumberlin, R. and Kaplan, R; NCI guidelines for IMRT as posted on ITC website (http://itc.wustl.edu; 05/17/04)http://itc.wustl.edu

7. What kind of accuracy is expected? Radiation needs to be Fullfilling Dose distribution across a volume Dose distribution against a standard –any discrepancy between the submitting institution's DVHs and those computed by the ITC in excess of +5% (or 3 cc for small structures) in total volume or +5% (relative to the absolute structure volume) of the volume calculated to be at or above the appropriate TD 5/5 dose for the particular structure will need to be resolved prior to successfully completing the Dry Run Test http://itc.wustl.edu/http://itc.wustl.edu/ 11/27/02 Image Guided Therapy Center

8. What kind of accuracy is required? Optimization must be better than good “In order to satisfy [the 1976 ICRU] recommendation [of 5%], each step in the radiotherapeutic process has to be performed at an accuracy better than 5%” Declich, F. et al, “dosimetric evaluation of a commercial 3-D treatment planning system using Report 55 by AAPM Task Group 23”Rad Onc 52: 69-77

9. What kind of accuracy is expected? When are outliers outliars? Target dose homogeneity –“Dose Variation within the Planning Target volume…should be kept within +7% and -5% of the prescribed dose…otherwise, it is the responsibility of the radiation oncologist to decide whether this can be accepted or not ” ICRU report 50, section 2.4.1.

10. Homogeneity and the Maximum dose Brush off the Biggest For 2D, max target dose only counted for an area >2cm 2 (ICRU Report 29) For 3D, max target dose only counted for a volume whose min. diameter is >15mm (ICRU 50) Outside the target, similarly sized volume called “hot spot” (ICRU 50)

11. Homogeneity and the minimum dose Small is dutiful “In contrast to the situation with the maximum absorbed dose, no volume is limit is recommended when reporting minimum dose.” (ICRU 50, section 2.4.4.)

12. Normal tissue maximums Can apply minimum volume rule if maximum dose is not critical Otherwise, a smaller volume is applied ICRU 50 Cord,e.g.

13. Normal Tissue Limits Normal Tissue and Tumor Dose Limits

14. Homogeneity Limit Objective Bound on Improvement 2.5% StructureVolumeHomogeneity limit (max – min dose) Target100% -5% ≤10%

15. Gaining Traction Sparsing of [a ij ] matrix Sampling of point set {i} Sampling of beamlet set {j} –Geometric (e.g, by target\normal tissue projections) –Random, iterative (fill in the holes) –Numerical – column generation

16. Sources of Discrepancy c 0.25 cm Report discrepancies - isodose plot grid ≠ dose calculation grid - pixel grid finer than dose calculation grid - dvh sampling is inferior - within voxel variations, if >50%/cm →>5% in 1mm 2 2 2 10 Optimization Failures ++ + + + +

17. Types of Discrepancies Violations –In volume –in Dose –In homogeneity Shortfalls

18. Dose (in Gy) % Volume Heart Volume 100% 50% 60 Gy35 Gy (1) Absolute dose limit 4% vol error 4% error in dose NEEDED: An Accounting for error in estimates of dose and volume when the dvh is constructed

19. Dose (in Gy) % Volume Heart Volume 100% 50% 60 Gy35 Gy (1) Absolute dose limit Type I reported violation: A dvh through this point would be clearly be in violation; it lies within the hatched region, that is placed outside the range of uncertainty in volume (4%) and dose (4%) 4% vol error 4% error in dose

20. Dose (in Gy) % Volume Heart Volume 100% 50% 60 Gy35 Gy (1) Absolute dose limit 4% vol error 4% error in dose Type II violation : A point lies outside the 4% dose calc error, but it could lie within the 4% uncertainty in estimating the volume holding that point

21. Dose (in Gy) % Volume Heart Volume 100% 50% 60 Gy35 Gy (1) Absolute dose limit Type III violation :The dose volume histogram lies within the region of ±4% uncertainty in dose and volume. But >4% volume exceeds the dose limit by some amount, and there is an error of >4% in dose within some part of the volume. The maximum recorded dose is plotted, along with the range of error in volume holding that dose (0%-4%)

22. Maximum violations in lung dose- volume limit in no shortfall 3D trials # sample points trials#trials with lung violations #trials with shortfalls Volume violation Dose violation 450351153%4.9 Gy 600351164%7.4 Gy 80034*882%4.1 Gy * one trial could not be solved Langer, M.; et al, “The reliability of optimization under dose- volume limits”, Int. J. Radiat Oncol. Biol. Phys. 26:529-538; 1993

23. Sampling and time Sampling points Avg time95 th percentile Maximum time 4503.78 min12.3 min.36 min. 6004.03 min17.9 min31 min 80056 min101 min1315 min Langer, M.; et al, “The reliability of optimization under dose-volume limits”, Int. J. Radiat Oncol. Biol. Phys. 26:529-538; 1993

24. Formulations and fabulations j=1 a i=1,j=1 j=2 a i=1,j=2 d i=1 Bahr, GK Radiol, 1968 Sonderman, Abrahamson, Op Res 1985 Langer, M, Int J Rad Onc 1990

25. l k g s=1 g s=2 segmentation

26. p s Rardin, R.; Langer, M.P.; Preiciado-Walters, F.; Thai, V. Column Generation for IMRT Cancer Therapy Optimization with Implementable Segments. Euro/Informs Joint International Meeting, #2704, July 6-10, 2003, Istanbul. (meeting presentation) Langer, M.; Thai, V. Delivery of Static Field IMRT is Improved by Avoiding Discretization of Intensity Levels. Int. J. Radiat. Oncol. Biol. Phys. 54(2):156; October 6-10, 2002, New Orleans, LA (meeting presentation). Langer, M.; Thai, V.; Papiez, L. Improved Leaf Sequencing Reduces Segments or Monitor Units Needed to Deliver IMRT Using Multileaf Collimators. Med. Phys. 28(12): 2450-2458; 2001. Preciado-Walters, F.; Langer, M.P.; Rardin, R.; Thai, V. Column generation for IMRT Cancer Therapy - Optimization with Implementable Segments.

27. Cord  45 Gy Target dose homogeneity .85 Maximize minimum target dose

28. Continuous Levels Discrete Levels Intensity map Continuous Levels relaxed homogeneity

29. StructureClinical plan Criteria IMRT constraint template (starting) PTVPTV 1 =54 Gy PTV 2 =70 Gy Max dose  120% prescribed PTV 1 prescribed=54Gy PTV 2 prescribed=70Gy PTV 1 =51.3-56.7 Gy PTV 2 =66.5-73.5 Gy Penalty =25 spinal cord Max dose  40 Gy Max dose = 35 Gy Penalty = 100 Brain stem Max dose  45 Gy Max dose = 35 Gy Penalty = 100 1.2 x (54)=64.8 1.2 x (70)=84 Hunt, MA et al, IJROBP 49:623-632; 2001

30. What is the effect of uncertainty in geometry?

31. Minimum Dose 79.2 79.6 79.8 McCormick, T.C.; Dink, D.; Orcun, S.; Pekny, J.; Rardin, R.; Baxter, L.; Langer, M. Projecting the Effect of Target Boundary Uncertainty on the Tumor Dose Prescription to Guide Contouring for Treatment Planning. Int. J. Radiat. Oncol.Biol. Phys. 57(2 suppl):S235 (2003) Amer. Soc. Therap. Radiol. Oncol. 45th Annual Meeting (meeting presentation).

32. Conclusions MDs can formulate the planning problems They are combinatorially complex We can identify levels of required accuracy Is there an advantage or disadvantage to incorporate delivery constraints?