Minimizing Setup and Beam-On Times in Radiation Therapy

Radiation therapy is one of the commonly used cancer therapies. The radiation treatment poses atuning problem: it needs to be effective enough to destroy the tumor, but it should maintain the func-tionality of the organs close to the tumor. Towards this goal the design of a radiation treatment has to becustomized for each patient. Part of this design are intensity matrices that define the radiation dosage ina discretization of the beam head. To minimize the treatment time of a patient the beam-on time and thesetup time need to be minimized. For a given row of the intensity matrix, the minimum beam-on timeis equivalent to the minimum number of binary vectors with the consecutive “1”s property that sum tothis row, and the minimum setup time is equivalent to the minimum number of distinct vectors in a setof binary vectors with the consecutive “1”s property that sum to this row. We give a simple linear timealgorithm to compute the minimum beam-on time. We prove that the minimum setup time problem isAPX-hard and give approximation algorithms for it using a duality property. For the general case, wegive a 2413 approximation algorithm. For unimodal rows, we give a 97 approximation algorithm. We alsoconsider other variants for which better approximation ratios exist. ∗IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, {nikhil,sbar}@us.ibm.com†IDA Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540, [email protected] work was done while at IBM T.J. Watson Research Center.