Intensity modulated radiotherapy treatment planning by use of a barrier-penalty multiplier method

The use of nonlinear functions describing biological effects has recently become a major goal in connection with intensity modulated radiotherapy planning models for cancer treatment. In this article, we present a new biological model for this purpose and discuss the solution of the related large-scale nonlinear optimization problems. The model includes equivalent uniform dose and partial volume constraints and employs tumor control probability as the objective. The resulting optimization problems are convex; there are nonconvex constrained optimization problems with several thousands of variables for which gradients of the involved functions are available, but the computation of Hessians is too costly. It is suggested to solve these problems using the barrier-penalty multiplier method by Polyak ((Polyak, R., 1992, Modified barrier functions (theory and methods). Mathematical Programming, 54, 177-222.], [Polyak, R., 2002, Nonlinear resealing vs. smoothing technique in convex optimization. Mathematical Programming, 92, 197-235.]) and Ben-Tal et al. and Ben-Tal and Zibulevsky ([Ben-Tal, A., Yuzefokich, I. and Zibulevsky, M., 1992, Penalty/barrier multiplier methods for minimax and constrained smooth convex problems. Technical Report 9/92, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Haifa, Israel.], [Ben-Tal, A. and Zibulevsky, M., 1997, Penalty/barrier multiplier methods for convex programming problems. SIAM Journal of Optimization, 7, 347-366.]), where this algorithm is modified according to ideas which are motivated by the related Lagrangian barrier algorithm of Conn et al. ([Conn, A.R., Gould, N.I.M. and Taint, P., 1992, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Maths, FUNDP, Namur, Belgium.], [Conn, A.R., Gould, N.I.M. and Taint, P.L., 1992, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Mathematics of Computation, 66,261-288.]). In particular, the subproblems in the algorithm are solved by a conjugate gradient method, as the spectrum of the Hessian of the Lagrangian at a solution of such problem indicates fast (local) convergence of the objective function values to a good approximate (locally) optimal value. Some characteristic numbers showing the average numerical performance of the algorithm are tabulated for various types of tumors and for a set of 127 clinical cases in total. Its capabilities and typical behavior also are illustrated explicitly by a computed therapy plan for a difficult clinical case of a larynx tumor.