1. Gamma Knife Surgery and Region Covering Aaron Epel

2. What is the Gamma Knife? Non-invasive cancer treatment Targeted gamma radiation Radiation “shots” radii of 2, 4, 7, and 9 mm

3. The problem Manually targeting collimators is time- consuming and not necessarily efficient Objective is to irradiate tumor Easy to damage surrounding healthy, sensitive brain tissue

4. The problem Maximize coverage of tumor region… …but minimize tissue damage and waste Must prioritize one or the other Related to problems of shape covering vs. shape packing

5. The problem Minimize the number of shots used, while: ▫Not irradiating any non-tumor tissue ▫Irradiating at least a certain percentage of tumor area

6. Approach 1: Wu and Bourland 1999 Assumptions ▫Doses of radiation in spherical or circular “shots” ▫Dose of radiation in a shot not uniform ▫Shots need not/may not overlap ▫Target tumor region is bounded, with known volume and surface ▫Four possible radii for shots, each equally available: 9mm, 7mm, 4mm, 2mm

7. Approach 1: Wu and Bourland 1999 The optimal arrangement of doses will: ▫Cover target region within a percentage tolerance ▫Minimize number of shots ▫Have all shots inside the region ▫Have no overlapping shots

8. Skeletonization Skeleton of an image: loci of centers of all circles tangent to at least 2 boundaries, contained entirely in the region Various algorithms may be used to find skeleton Similar to medial axis transform

9. Approach 1: Wu and Bourland 1999 Skeletonization approach ▫If optimal arrangement exists, all shots have center on some sub-region’s skeleton

10. Approach 1: Wu and Bourland 1999 Iterated method ▫1)Make a skeleton for the tumor region ▫2)Find all potential shots for each:  End point  Cross point  Point where shot is tangent to region boundaries ▫3)For each potential shot:  Redraw the region with that shot’s area deleted  Make a new skeleton for the sub-region  Repeat 2) and 3) until area covered > tolerance threshold

11. Iterated method example

12. An example: triangular region

14. This arrangement had the greatest ratio of area covered to tumor area: 72%, but still much less than 90% This is due to the region size

15. Extensions Simulated annealing (Zhang et al 2003) ▫Initialized using similar process to above ▫Random walks for shot location, then radius ▫Allows overlap and spill over tumor boundaries Similar method for another formulation? ▫Minimize the excess irradiation to the patient and the number of shots, while covering entire tumor ▫“Cost” is a function of healthy tissue area irradiated and number of shots

16. Conclusions Fundamentally different formulations of the problem ▫Tradeoff: effectiveness vs. limit on damage ▫Related to circle covering vs. circle packing Extension to three dimensions for application Image based: applies to irregular regions Beneficial in determining treatment plans

17. References Fisher, R, Perkins, S, Walker, A, and Wolfart, E. “Skeletonization/Medial axis transform.” 2004. Friedman, Erich. “Erich’s packing center.” 2009. Nurmela, Kari J. “Conjecturally optimal coverings of an equilateral triangle with up to 36 circles.” Experimental Mathematics (2000)9.2:241-250. Palágyi, K. “Skeletonization.” 2001. Wu, Q. and Bourland, J. “Morphology-guided radiosurgery treatment planning and optimization for multiple isocenters.” Med. Phys. (1999)26.10:2151-2160. Zhang, Pengpeng et al. “Plug pattern optimization for gamma knife radiosurgery treatment planning.” Intl. J. Radiation Oncology Biol. Phys. (2003)55.2:420-427. “Gamma Knife® Surgery.” IRSA. 2009.