Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem

Optimization is of vital importance when performing intensity modulated radiation therapy to treat cancer tumors. The optimization problem is typically large-scale with a nonlinear objective function and bounds on the variables, and we solve it using a quasi-Newton sequential quadratic programming method. This study investigates the effect on the optimal solution, and hence treatment outcome, when solving an approximate optimization problem of lower dimension. Through a spectral decomposition, eigenvectors and eigenvalues of an approximation to the Hessian are computed. An approximate optimization problem of reduced dimension is formulated by introducing eigenvector weights as optimization parameters, where only eigenvectors corresponding to large eigenvalues are included. The approach is evaluated on a clinical prostate case. Compared to bixel weight optimization, eigenvector weight optimization with few parameters results in faster initial decline in the objective function, but with inferior final solution. Another approach, which combines eigenvector weights and bixel weights as variables, gives lower final objective values than bixel weight optimization does. However, this advantage comes at the expense of the pre-computational time for the spectral decomposition.