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Covering problems arising in Gamma Knife radiosurgery

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A widespread technology for curing brain tumours is the so-called “Gamma Knife”. This relies on irradiating the tumour with gamma radiation so as to kill cancerous cells. The problem is that the 3D shape of tumours might be arbitrary, but the gamma radiation hits a spherical volume. Radiation intensity depends on the sphere radius and also on the length of irradiation. The gamma knife machine requires a parametrization in both senses: choice of spherical radii, positions of sphere centers, and choice of irradiation time. Optimally deciding the value of these parameters requires the solution of a complex Mixed-Integer Nonlinear Problem. We look at the sphere covering subproblem, and suggest some possible solution methods for regularly (but, alas, irrealistically) shaped tumours.

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