Approximated matrix decomposition for IMRT planning with multileaf collimators

A standard problem in intensity modulated radiation therapy is the representation of a given intensity matrix, i.e. a matrix of nonnegative integers, as a nonnegative linear combination of special 0-1-matrices, called segments. These segments can be practically realized by multileaf collimators. One important aim is the minimization of the sum of the coefficients of the linear combination, i.e. the delivery time. In this article, we study the question how much the delivery time can be reduced if some small deviation from the given intensity matrix is allowed. We characterize the optimal solutions for one-row matrices and show that the approximation can be carried out in an iterative way. The structural characterization yields a fast algorithm that minimizes the delivery time and then also the deviation. Moreover, algorithms for the general case together with numerical results are presented.