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Utilizing Problem Structure in Optimization of Radiation Therapy
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
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 [en]
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: urn:nbn:se:kth:diva-4689ISBN: 978-91-7178-920-4 (print)OAI: oai:DiVA.org:kth-4689DiVA: diva2:13441
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
List of papers
1. Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem
Open this publication in new window or tab >>Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem
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.

Keyword
IMRT; optimization; sequential quadratic programming; quasi-Newton method; MULTIPLE LOCAL MINIMA; RADIOTHERAPY
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8183 (URN)10.1007/s10479-006-0082-z (DOI)000242603100006 ()2-s2.0-33845333863 (Scopus ID)
Note
QC 20100709Available from: 2008-04-03 Created: 2008-04-03 Last updated: 2017-12-14Bibliographically approved
2. Iterative regularization in intensity-modulated radiation therapy optimization
Open this publication in new window or tab >>Iterative regularization in intensity-modulated radiation therapy optimization
2006 (English)In: Medical physics (Lancaster), ISSN 0094-2405, Vol. 33, no 1, 225-234 p.Article in journal (Refereed) Published
Abstract [en]

A common way to solve intensity-modulated radiation therapy (IMRT) optimization problems is to use a beamlet-based approach. The approach is usually employed in a three-step manner: first a beamlet-weight optimization problem is solved, then the fluence profiles are converted into stepand-shoot segments, and finally postoptimization of the segment weights is performed. A drawback of beamlet-based approaches is that beamlet-weight optimization problems are ill-conditioned and have to be regularized in order to produce smooth fluence profiles that are suitable for conversion. The purpose of this paper is twofold: first, to explain the suitability of solving beamlet-based IMRT problems by a BFGS quasi-Newton sequential quadratic programming method with diagonal initial Hessian estimate, and second, to empirically show that beamlet-weight optimization problems should be solved in relatively few iterations when using this optimization method. The explanation of the suitability is based on viewing the optimization method as an iterative regularization method. In iterative regularization, the optimization problem is solved approximately by iterating long enough to obtain a solution close to the optimal one, but terminating before too much noise occurs. Iterative regularization requires an optimization method that initially proceeds in smooth directions and makes rapid initial progress. Solving ten beamlet-based IMRT problems with dose-volume objectives and bounds on the beamlet-weights, we find that the considered optimization method fulfills the requirements for performing iterative regularization. After segment-weight optimization, the treatments obtained using 35 beamlet-weight iterations outperform the treatments obtained using 100 beamlet-weight iterations, both in terms of objective value and of target uniformity. We conclude that iterating too long may in fact deteriorate the quality of the deliverable plan.

Keyword
intensity-modulated radiation therapy; quasi-Newton method; conjugate gradient method; regularization; iterative regularization; IMAGE-RECONSTRUCTION; IMRT OPTIMIZATION; PHOTON BEAMS; RADIOTHERAPY; DELIVERY; CONSTRAINTS; DEGENERACY; ALGORITHMS; EFFICIENCY; PATTERNS
National Category
Computational Mathematics Radiology, Nuclear Medicine and Medical Imaging
Identifiers
urn:nbn:se:kth:diva-8184 (URN)10.1118/1.2148918 (DOI)000234772100028 ()2-s2.0-31044450092 (Scopus ID)
Note
QC 20100709Available from: 2008-04-03 Created: 2008-04-03 Last updated: 2017-12-14Bibliographically approved
3. Combining segment generation with direct step-and-shoot optimization in intensity-modulated radiation therapy
Open this publication in new window or tab >>Combining segment generation with direct step-and-shoot optimization in intensity-modulated radiation therapy
2008 (English)In: Medical physics (Lancaster), ISSN 0094-2405, Vol. 35, no 9, 3828-3838 p.Article in journal (Refereed) Published
Abstract [en]

A method for generating a sequence of intensity-modulated radiation therapy step-and-shoot plans with increasing number of segments is presented. The objectives are to generate high-quality plans with few, large and regular segments, and to make the planning process more intuitive. The proposed method combines segment generation with direct step-and-shoot optimization, where leaf positions and segment weights are optimized simultaneously. The segment generation is based on a column generation approach. The method is evaluated on a test suite consisting of five head-and-neck cases and five prostate cases, planned for delivery with an Elekta SLi accelerator. The adjustment of segment shapes by direct step-and-shoot optimization improves the plan quality compared to using fixed segment shapes. The improvement in plan quality when adding segments is larger for plans with few segments. Eventually, adding more segments contributes very little to the plan quality, but increases the plan complexity. Thus, the method provides a tool for controlling the number of segments and, indirectly, the delivery time. This can support the planner in finding a sound trade-off between plan quality and treatment complexity.

Keyword
intensity-modulated radiation therapy; step-and-shoot delivery; optimization; column generation; DIRECT-APERTURE OPTIMIZATION; COLUMN GENERATION; IMRT; RADIOTHERAPY; NUMBER; TIME
National Category
Computational Mathematics Radiology, Nuclear Medicine and Medical Imaging
Identifiers
urn:nbn:se:kth:diva-8185 (URN)10.1118/1.2964096 (DOI)000258773000002 ()2-s2.0-50449095767 (Scopus ID)
Note
QC 20100709. Uppdaterad från in press till published (20100709).Available from: 2008-04-03 Created: 2008-04-03 Last updated: 2017-12-14Bibliographically approved
4. A conjugate-gradient based approach for approximate solutions of quadratic programs
Open this publication in new window or tab >>A conjugate-gradient based approach for approximate solutions of quadratic programs
(English)In: Annals of Operations Research, ISSN 0254-5330, E-ISSN 1572-9338Article in journal (Refereed) Submitted
Abstract [en]

This paper deals with numerical behaviour and convergence properties of a recently presented column generation approach for optimization of so called step-and-shoot radiotherapy treatment plans. The approach and variants of it have been reported to be efficient in practice, finding near-optimal solutions by generating only a low number of columns. The impact of different restrictions on the columns in a column generation method is studied, and numerical results are given for quadratic programs corresponding to three patient cases. In particular, it is noted that with a bound on the two-norm of the columns, the method is equivalent to the conjugate-gradient method. Further, the above-mentioned column generation approach for radiotherapy is obtained by employing a restriction based on the infinity-norm and non-negativity. The column generation method has weak convergence properties if restricted to generating feasible step-and-shoot plans, with a "tailing-off" effect for the objective values. However, the numerical results demonstrate that, like the conjugate-gradient method, a rapid decrease of the objective value is obtained in the first few iterations. For the three patient cases, the restriction on the columns to generate feasible step-and-shoot plans has small effect on the numerical efficiency.

Keyword
column generation; conjugate-gradient method; intensity-modulated radiation therapy; step-and-shoot delivery
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8186 (URN)
Note
QC 20100709Available from: 2008-04-03 Created: 2008-04-03 Last updated: 2017-12-14Bibliographically approved

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