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Mathematical Optimization of HDR Brachytherapy
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.ORCID iD: 0000-0003-2220-6125
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

One out of eight deaths throughout the world is due to cancer. Developing new treatments and improving existing treatments is hence of major importance. In this thesis we have studied how mathematical optimization can be used to improve an existing treatment method: high-dose-rate (HDR) brachytherapy.

HDR brachytherapy is a radiation modality used to treat tumours of for example the cervix, prostate, breasts, and skin. In HDR brachytherapy catheters are implanted into or close to the tumour volume. A radioactive source is moved through the catheters, and by adjusting where the catheters are placed, called catheter positioning, and how the source is moved through the catheters, called the dwelling time pattern, the dose distribution can be controlled.

By constructing an individualized catheter positioning and dwelling time pattern, called dose plan, based on each patient's anatomy, it is possible to improve the treatment result. Mathematical optimization has during the last decade been used to aid in creating individualized dose plans. The dominating optimization model for this purpose is a linear penalty model. This model only considers the dwelling time pattern within already implanted catheters, and minimizes a weighted deviation from dose intervals prescribed by a physician.

In this thesis we show that the distribution of the basic variables in the linear penalty model implies that only dwelling time patterns that have certain characteristics can be optimal. These characteristics cause troublesome inhomogeneities in the plans, and although various measures for mitigating these are already available, it is of fundamental interest to understand their cause.

We have also shown that the relationship between the objective function of the linear penalty model and the measures commonly used for evaluating the quality of the dose distribution is weak. This implies that even if the model is solved to optimality there is no guarantee that the generated plan is optimal with respect to clinically relevant objectives, or even near-optimal. We have therefore constructed a new model for optimizing the dwelling time pattern. This model approximates the quality measures by the concept conditional value-at-risk, and we show that the relationship between our new model and the quality measures is strong. Furthermore, the new model generates dwelling time patterns that yield high-quality dose distributions.

Combining optimization of the dwelling time pattern with optimization of the catheter positioning yields a problem for which it is rarely possible to find a proven optimal solution within a reasonable time frame. We have therefore developed a variable neighbourhood search heuristic that outperforms a state-of-the-art optimization software (CPLEX). We have also developed a tailored branch-and-bound algorithm that is better at improving the dual bound than a general branch-and-bound algorithm. This is a step towards the development of a method that can find proven optimal solutions to the combined problem within a reasonable time frame.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1550
Natural Sciences
##### Identifiers
ISBN: 978-91-7519-496-7 (print)OAI: oai:DiVA.org:liu-99795DiVA, id: diva2:658212
##### Public defence
2013-11-28, Nobel (BL32), B-huset, ingång 23, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
##### Supervisors
Available from: 2013-11-05 Created: 2013-10-21 Last updated: 2013-11-05Bibliographically approved
##### List of papers
1. Impact of Using Linear Optimization Models in Dose Planning for HDR Brachytherapy
Open this publication in new window or tab >>Impact of Using Linear Optimization Models in Dose Planning for HDR Brachytherapy
2012 (English)In: Medical physics (Lancaster), ISSN 0094-2405, Vol. 39, no 2, p. 1021-1028Article in journal (Refereed) Published
##### Abstract [en]

Purpose: Dose plans generated with optimization models hitherto used in HDR brachytherapy have shown a tendency to yield longer dwell times than manually optimized plans. Concern has been raised for the corresponding undesired hot spots and various methods to mitigate these have been developed. The hypotheses of this work are a) that one cause for the long dwell times is the use of objective functions comprising simple linear penalties and b) that alternative penalties, being piecewise linear, would lead to reduced length of individual dwell times.

Methods: The characteristics of the linear penalties and the piecewise linear penalties are analysed mathematically. Experimental comparisons between the two types of penalties are carried out retrospectively for a set of prostate cancer patients.

Results: While most dose-volume parameters do not differ significantly between the two types of penalties significant changes can be seen in the dwell times. On the average, total dwell times were reduced by 4.2%, with a reduction of maximum dwell times by 30%, using the alternative penalties.

Conclusion: The use of linear penalties in optimization models for HDR brachytherapy is one cause for undesired longer dwell times appearing in mathematically optimized plans. By introducing alternative penalties significant reduction in dwell times can be achieved for HDR brachytherapy dose plans. Although various constraints as to reduce the long dwell times have been developed our finding is of fundamental interest in showing the shape of the objective function to be one reason for their appearance.

##### Place, publisher, year, edition, pages
American Association of Physicists in Medicine, 2012
##### Keyword
Brachytherapy, Optimization, Treatment planning, Linear programming, Piecewise linear functions
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-67786 (URN)10.1118/1.3676179 (DOI)000300215800048 ()
Available from: 2011-04-26 Created: 2011-04-26 Last updated: 2017-12-11Bibliographically approved
2. Study of the Relationship Between Dosimetric Indices and Linear Penalties in Dose Distribution Optimization for HDR Prostate Brachytherapy
Open this publication in new window or tab >>Study of the Relationship Between Dosimetric Indices and Linear Penalties in Dose Distribution Optimization for HDR Prostate Brachytherapy
##### Abstract [en]

Purpose: Most clinical software for optimizing dwelling time patterns is based on a linear penalty model. The quality of a dose distribution generated by the dwelling time pattern is, however, evaluated through a number of dosimetric indices. The purpose of this article is to investigate the relationship between the linear penalty model and the dosimetric indices.

Method and Materials: Data sets from three patients, previously treated for prostate cancer with HDR brachytherapy as a boost to external beam therapy, were used for this study, and for each of them 300 random dwelling time patterns were generated. The relationship between the linear penalty model and the dosimetric indices were studied both by the Pearson’s product moment correlation coefficient between the objective function value of the linear penalty model and the values of the dosimetric indices, and by scatter-grams.

Results: For one of the three patients we found a clear connection between the linear penalty model and the values of the dosimetric indices, but not for the other two. For the two patients without a clear connection there where some dosimetric indices that actually improved with deteriorating objective function value.

Conclusion: The dwelling time pattern found by using the linear penalty model does not correspond to the optimal dose distribution with respect to dosimetric indices.

##### Keyword
Optimization, Linear penalty models, Correlation, HDR Brachytherapy
##### National Category
Medical and Health Sciences
##### Identifiers
urn:nbn:se:liu:diva-100388 (URN)
Available from: 2013-11-05 Created: 2013-11-05 Last updated: 2013-11-05Bibliographically approved
3. A linear programming model for optimizing HDR brachytherapy dose distributions with respect to mean dose in the DVH-tail
Open this publication in new window or tab >>A linear programming model for optimizing HDR brachytherapy dose distributions with respect to mean dose in the DVH-tail
2013 (English)In: Medical physics (Lancaster), ISSN 0094-2405, Vol. 40, no 8Article in journal (Refereed) Published
##### Abstract [en]

Purpose: Recent research has shown that the optimization model hitherto used in high-dose-rate (HDR) brachytherapy corresponds weakly to the dosimetric indices used to evaluate the quality of a dose distribution. Although alternative models that explicitly include such dosimetric indices have been presented, the inclusion of the dosimetric indices explicitly yields intractable models. The purpose of this paper is to develop a model for optimizing dosimetric indices that is easier to solve than those proposed earlier. less thanbrgreater than less thanbrgreater thanMethods: In this paper, the authors present an alternative approach for optimizing dose distributions for HDR brachytherapy where dosimetric indices are taken into account through surrogates based on the conditional value-at-risk concept. This yields a linear optimization model that is easy to solve, and has the advantage that the constraints are easy to interpret and modify to obtain satisfactory dose distributions. less thanbrgreater than less thanbrgreater thanResults: The authors show by experimental comparisons, carried out retrospectively for a set of prostate cancer patients, that their proposed model corresponds well with constraining dosimetric indices. All modifications of the parameters in the authors model yield the expected result. The dose distributions generated are also comparable to those generated by the standard model with respect to the dosimetric indices that are used for evaluating quality. less thanbrgreater than less thanbrgreater thanConclusions: The authors new model is a viable surrogate to optimizing dosimetric indices and quickly and easily yields high quality dose distributions.

##### Place, publisher, year, edition, pages
American Association of Physicists in Medicine, 2013
##### Keyword
brachytherapy, optimization, treatment planning, dosimetric indices, conditional value-at-risk
##### National Category
Medical and Health Sciences
##### Identifiers
urn:nbn:se:liu:diva-97239 (URN)10.1118/1.4812677 (DOI)000322735900010 ()
##### Note

Funding Agencies|Swedish Cancer Society (Cancerfonden)|100512|

Available from: 2013-09-05 Created: 2013-09-05 Last updated: 2017-12-06
4. Heuristics for Integrated Optimization of Catheter Positioning and Dwell Time Distribution in Prostate HDR Brachytherapy
Open this publication in new window or tab >>Heuristics for Integrated Optimization of Catheter Positioning and Dwell Time Distribution in Prostate HDR Brachytherapy
2016 (English)In: Annals of Operations Research, ISSN 0254-5330, E-ISSN 1572-9338, Vol. 236, no 2, p. 319-339Article in journal (Refereed) Published
##### Abstract [en]

High dose-rate (HDR) brachytherapy is a kind of radiotherapy used to treat, among others, prostate cancer. When applied to prostate cancer a radioactive source is moved through catheters implanted into the prostate. For each patient a treatment plan is constructed that decide for example catheter placement and dwell time distribution, that is where to stop the radioactive source and for how long.

Mathematical optimization methods has been used to find quality plans with respect to dwell time distribution, however few optimization approaches regarding catheter placement have been studied. In this article we present an integrated optimization model that optimize catheter placement and dwell time distribution simultaneously. Our results show that integrating the two decisions yields greatly improved plans, from 15% to 94% improvement.

Since the presented model is computationally demanding to solve we also present three heuristics: tabu search, variable neighbourhood search and genetic algorithm. Of these variable neighbourhood search is clearly the best, outperforming a state-of-the-art optimization software (CPLEX) and the two other heuristics.

Springer, 2016
##### Keyword
Brachytherapy, Dose planning, Catheter positioning, Mixed integer programming, Metaheuristics
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-67788 (URN)10.1007/s10479-013-1448-7 (DOI)000368946400003 ()
Available from: 2011-04-26 Created: 2011-04-26 Last updated: 2017-12-11Bibliographically approved
5. A Tailored Branch-and-Bound Method for Optimizing the Dwelling Time Pattern and Catheter Positioning in HDR Brachytherapy
Open this publication in new window or tab >>A Tailored Branch-and-Bound Method for Optimizing the Dwelling Time Pattern and Catheter Positioning in HDR Brachytherapy
##### Abstract [en]

High dose-rate (HDR) brachytherapy is one type of treatment for prostate cancer, in which a radioactive source is moved through catheters implanted into the prostate. For each patient, a unique treatment plan is constructed. This plan determines for example the catheter positioning and the dwelling time pattern, that is, where and for how long the source should stop.

Mathematical optimization methods are frequently used to find high-quality dwelling time patterns. However, choosing the catheter positioning is usually done without any aid of mathematical optimization methods. Researchers have recently suggested some optimization models for catheter positioning, and also heuristics for solving them. However, there are no available methods for finding the optimal solution of these models within a clinically acceptable time frame.

In this paper we present the foundation for a branch-and-bound method that has been tailored to the catheter positioning problem. Tests show that this tailored branch-and-bound method has some promising features, for example that the dual bound is improved faster than when using a standard branch-and-bound method. But the tests also show that further research is required to develop it into a method that can find the optimal solution fast enough.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 2013:12
##### Keyword
Branch-and-Bound, Branching rules, Brachytherapy, Dose planning, Catheter positioning
##### National Category
Other Mathematics
##### Identifiers
urn:nbn:se:liu:diva-99784 (URN)LiTH-MAT-R– 2013/12–SE (ISRN)
Available from: 2013-10-21 Created: 2013-10-21 Last updated: 2013-11-06Bibliographically approved

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Cite
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