On the degeneracy of the IMRT optimization problem

Abstract

One approach to the computation of photon IMRT treatment plans is the formulation of an optimization problem with an objective function that derives from an objective density. An investigation of the second-order properties of such an objective function in a neighborhood of the minimizer opens an intuitive access to many traits of this approach. A general finding is that only a small subset of the parameter space has nonzero curvature, while the objective function is entirely flat in a neighborhood of the minimizer in most directions. The dimension of the subspace of vanishing curvature serves as a measure for the degeneracy of the solution. This finding is important both for algorithm design and evaluation of the mathematical model of clinical intuition, expressed by the objective function. The structure of the subspace of great curvature is found to be imposed on the problem by conflicts between objectives of target and critical structures. These conflicts and their corresponding modes of resolution form a common trait between all reasonable treatment plans of a given case. The high degree of degeneracy makes the use of a conjugate gradient optimization algorithm particularly favorable since the number of iterations to convergence is equivalent to the number of different eigenvalues of the curvature tensor and is hence independent from the number of optimization parameters. A high level of degeneracy of the fluence profiles implies that it should be possible to stipulate further delivery-related conditions without causing severe deterioration of the dose distribution.

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

“…This redundancy makes the P T P matrix degenerate. It was observed in Alber et al (2002) that the Hessian often has few significant eigenvalues at the optimum and that the Hessian at optimum is more degenerate than P T P itself. The objective function thus increases the amount of degeneracy of the problem.…”

“…Such an approach was introduced in Markman et al (2002). The motivation for reducing the problem dimension is that the problem has been observed to be degenerate in the sense that the Hessian of the objective function has a large number of small eigenvalues and rather few large eigenvalues (Alber et al, 2002).…”

Reduced order constrained optimization (ROCO): Clinical application to lung IMRT

“…Then we conjecture that if we randomly choose parameter sets P q in the neighborhood of P opt , the resulting I q from unconstrained optimization will define a small basis which spans a space containing such an I opt . 26 We choose this neighborhood from clinical experience to include the range of values that planners have used for similar cases. Once a range of parameters has been chosen, Latin hypercube sampling is used to choose N samp parameter sets at which to sample; Latin hypercube sampling is a particular case of stratified sampling that achieves an efficient coverage of the space of input parameters.…”

“…However, the clinical plan should not be viewed as a "ground truth" correct answer; several authors have noted a high degree of degeneracy in IMRT plans, which result in similar objective function values but different clinical tradeoffs. 26 We conjecture that the degree of this degeneracy is greater for lung patients than for prostate patients, because our PCA modes are not able to represent the clinical plan as well: the value of the projection residual R [see Eq. (8) and Fig.…”

A Two-Stage Programming Approach to Fluence Map Optimization for Intensity-Modulated Radiation Therapy Treatment Planning

“…After an acceptable set of fluence maps is produced, one must find a suitable way for delivery (realization problem). Typically, beamlet intensities are discretized and one of the many existing techniques ( [3,11]) is used to construct the apertures and intensities that approximately match the intensity maps previously determined. However, plan's quality deterioration must be prevented when reproducing the optimized intensity maps [12,13].…”