Gamma Knife Surgery and Region Covering Aaron Epel.
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Transcript of Gamma Knife Surgery and Region Covering Aaron Epel.
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Gamma Knife Surgeryand Region CoveringAaron Epel
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What is the Gamma Knife?
•Non-invasive cancer treatment•Targeted gamma radiation•Radiation “shots” radii of 2, 4, 7, and 9 mm
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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
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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
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The problem
•Minimize the number of shots used, while:▫ Not irradiating any non-tumor tissue▫ Irradiating at least a certain percentage of tumor area
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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
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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
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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
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Approach 1: Wu and Bourland 1999•Skeletonization approach▫If optimal arrangement exists, all shots have center on some sub-region’s skeleton
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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
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Iterated method example
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An example: triangular region
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An example: triangular region
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An example: triangular region
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
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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
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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
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References• Fisher, R, Perkins, S, Walker, A, and Wolfart, E. “Skeletonization/Medial axis transform.” 2004. <http://homepages.inf.ed.ac.uk/rbf/HIPR2/skeleton.htm>
• Friedman, Erich. “Erich’s packing center.” 2009. <http://www2.stetson.edu/~efriedma/packing.html>
• 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. <http://www.inf.u-szeged.hu/~palagyi/skel/skel.html>
• 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. <http://www.irsa.org/gamma_knife.html>