Coupled Path Planning, Region Optimization, and Applications in Intensity-modulated Radiation Therapy

In this paper, we consider an optimization problem in discrete geometry, called coupled path planning (CPP). Given a finite rectangular grid and a non-negative function f defined on the horizontal axis of the grid, we seek two non-crossing monotone paths in the grid, such that the vertical difference between the two paths approximates f in the best possible way. This problem arises in intensity-modulated radiation therapy (IMRT), where f represents an ideal radiation dose distribution and the two coupled paths represent the motion trajectories (or control sequence) of two opposing metal leaves of a delivery device for controlling the area exposed to the radiation source. By finding an optimal control sequence, the CPP problem aims to deliver precisely a prescribed radiation dose, while minimizing the side-effects on the surrounding normal tissue. We present efficient algorithms for different versions of the CPP problems. Our results are based on several new ideas and geometric observations, and substantially improve the solutions based on standard techniques. Implementation results show that our CPP algorithms run fast and produce better quality clinical treatment plans than the previous methods.