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Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.ORCID iD: 0000-0002-6252-7815
RaySearch Labs.
RaySearch Labs.
2006 (English)In: Annals of Operations Research, ISSN 0254-5330, E-ISSN 1572-9338, Vol. 148, no 1, 81-94 p.Article in journal (Refereed) Published
Abstract [en]

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 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.

Place, publisher, year, edition, pages
2006. Vol. 148, no 1, 81-94 p.
Keyword [en]
IMRT; optimization; sequential quadratic programming; quasi-Newton method; MULTIPLE LOCAL MINIMA; RADIOTHERAPY
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-8183DOI: 10.1007/s10479-006-0082-zISI: 000242603100006Scopus ID: 2-s2.0-33845333863OAI: oai:DiVA.org:kth-8183DiVA: diva2:13437
Note
QC 20100709Available from: 2008-04-03 Created: 2008-04-03 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Utilizing Problem Structure in Optimization of Radiation Therapy
Open this publication in new window or tab >>Utilizing Problem Structure in Optimization of Radiation Therapy
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

In this thesis, optimization approaches for intensity-modulated radiation therapy are developed and evaluated with focus on numerical efficiency and treatment delivery aspects. The first two papers deal with strategies for solving fluence map optimization problems efficiently while avoiding solutions with jagged fluence profiles. The last two papers concern optimization of step-and-shoot parameters with emphasis on generating treatment plans that can be delivered efficiently and accurately. In the first paper, the problem dimension of a fluence map optimization problem is reduced through a spectral decomposition of the Hessian of the objective function. The weights of the eigenvectors corresponding to the p largest eigenvalues are introduced as optimization variables, and the impact on the solution of varying p is studied. Including only a few eigenvector weights results in faster initial decrease of the objective value, but with an inferior solution, compared to optimization of the bixel weights. An approach combining eigenvector weights and bixel weights produces improved solutions, but at the expense of the pre-computational time for the spectral decomposition. So-called iterative regularization is performed on fluence map optimization problems in the second paper. The idea is to find regular solutions by utilizing an optimization method that is able to find near-optimal solutions with non-jagged fluence profiles in few iterations. The suitability of a quasi-Newton sequential quadratic programming method is demonstrated by comparing the treatment quality of deliverable step-and-shoot plans, generated through leaf sequencing with a fixed number of segments, for different number of bixel-weight iterations. A conclusion is that over-optimization of the fluence map optimization problem prior to leaf sequencing should be avoided. An approach for dynamically generating multileaf collimator segments using a column generation approach combined with optimization of segment shapes and weights is presented in the third paper. Numerical results demonstrate that the adjustment of leaf positions improves the plan quality and that satisfactory treatment plans are found with few segments. The method provides a tool for exploring the trade-off between plan quality and treatment complexity by generating a sequence of deliverable plans of increasing quality. The final paper is devoted to understanding the ability of the column generation approach in the third paper to find near-optimal solutions with very few columns compared to the problem dimension. The impact of different restrictions on the generated columns is studied, both in terms of numerical behaviour and convergence properties. A bound on the two-norm of the columns results in the conjugate-gradient method. Numerical results indicate that the appealing properties of the conjugate-gradient method on ill-conditioned problems are inherited in the column generation approach of the third paper.

Place, publisher, year, edition, pages
Stockholm: KTH, 2008. xi, 32 p.
Series
Trita-MAT. OS, ISSN 1401-2294 ; 08/OS/03
Keyword
Optimization, intensity-modulated radiation therapy, conjugate-gradient method, step-and-shoot delivery, column generation, quasi-Newton method, regularization, sequential quadratic programming
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-4689 (URN)978-91-7178-920-4 (ISBN)
Public defence
2008-04-25, F3, Lindstedtsvägen 26, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20100709Available from: 2008-04-03 Created: 2008-04-03 Last updated: 2010-07-09Bibliographically approved

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