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

Abstract

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 quasiNewton 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 decompostion, 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 what bixel weight optimization does. However, this advantage comes at the expense of the pre-computational time for the spectral decomposition.

Combining segment generation with direct step‐and‐shoot optimization in intensity‐modulated radiation therapy

“…In our study, the objective function F is a weighted sum of standard IMRT penalty functions such as one-and two-sided quadratic penalty functions and gEU D, see e.g. [7] for mathematical formulations of the former and [17] for the latter. The parameters of the penalty functions are set up according to RTOG protocols; see Section 3.2 and Table 1 for details.…”

Reduced-order constrained optimization in IMRT planning

“…Also, the eigenvectors for the large eigenvalues span the subspace of I where F ( I ) changes rapidly, which corresponds to difficult trade-offs between targets and OARs. Carlsson et al (2006) further parameterized the intensities I by a few dominant eigenvectors from the Hessian, i.e., I = V p ξ, which is the to (5), the same as (9) except that V p contains p dominant eigenvectors from H . According Hessian in (10) can be written in the matrix form: $$H=A^{T}DA,D=\mathrm{diag}true\{\frac{1}{N},\cdots ,\frac{1}{N}{w}_{\min }ϴ(D_{\min }-D_i),\cdots ,\frac{1}{N}{w}_{\max }ϴ(D_j-D_{\max }),\cdots true\}$$ which depends on the dose coefficients A ij , the current dose distribution D i and the parameter settings ( w min , w max , etc).…”

Reduced-order Constrained Optimization in IMRT Planning

“…The IMRT degeneracy problem (i.e., the phenomenon that different intensity distributions can lead to nearly identical dose statistics) is related to the small eigenvalues of H. Also, the eigenvectors for the large eigenvalues span the subspace of I where F(I) changes rapidly, which corresponds to difficult trade-offs between targets and OARs. Carlsson et al (2006) further parameterized the intensities I by a few dominant eigenvectors from the Hessian, i.e., I = V p ξ, which is the to (5), the same as (9) except that V p contains p dominant eigenvectors from H. According Hessian in (10) can be written in the matrix form:…”