Computing tumour coverage as a result of respiratory motion during radiotherapy treatment

The objectives of radiotherapy treatment is to kill cancerous cells while minimizing damage to surrounding healthy tissues. The tumour location uncertainty "forces" oncologists to prescribe a larger treatment area than required in order to ensure that the whole tumour is receiving the prescribed dose. The problem is more acute when a tumour can move during treatment, e.g., as a result of breathing. In this paper, we present an algorithm for computing the area covered by a tumor as a result of a cyclic motion during treatment. Our algorithm solves the following geometric problem: Given an n-vertices convex polygon P = {v 1 v 2,... v n}, a monotone chain C = {c 1, c 2..., c m}, compute a minimums area polygon Q that includes all the space covered by P as it is translated along C such that v 1 ∈ P touches C. Here, we present a simple algorithm when P is a convex polygon. Our algorithm takes O(mn + m log n log(m + n)) time in the worst case.