The design of an intensity modulated radiotherapy treatment includes the selection of beam angles (geometry problem), the computation of an intensity map for each selected beam angle (intensity problem), and finding a sequence of configurations of a multileaf collimator to deliver the treatment (realization problem). Until the end of the last century research on radiotherapy treatment design has been published almost exclusively in the medical physics literature. However, since then, the attention of researchers in mathematical optimization has been drawn to the area and important progress has been made. In this paper we survey the use of optimization models, methods, and theories in intensity modulated radiotherapy treatment design. This is an updated version of the paper that appeared in 4OR, 6(3), 199-262 (2008).

The design of an intensity modulated radiotherapy treatment includes the selection of beam angles (geometry problem), the computation of an intensity map for each selected beam angle (intensity problem), and finding a sequence of configurations of a multileaf collimator to deliver the treatment (realization problem). Until the end of the last century research on radiotherapy treatment design has been published almost exclusively in the medical physics literature. However, since then, the attention of researchers in mathematical optimization has been drawn to the area and important progress has been made. In this paper we survey the use of optimization models, methods, and theories in intensity modulated radiotherapy treatment design.H.W. Hamacher and Ç.

Given a directed graph G = (N, A) with arc capacities u ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u for the arc set A such that a given feasible flowx is optimal with respect to the modified capacities. Among all capacity vectorsû satisfying this condition, we would like to find one with minimum û − u value.We consider two distance measures for û − u , rectilinear (L 1 ) and Chebyshev (L ∞ ) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is N P-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.