1 / 41

Abstract

Surprise Functions and Optimization Under Uncertainty: Methods and Applications in Radiation Therapy Math Clinic – Mathematics of Medical Imaging and Radiation Therapy Fall 2002 Weldon A. Lodwick. Abstract.

derron
Download Presentation

Abstract

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Surprise Functions and Optimization Under Uncertainty: Methods and Applications in Radiation TherapyMath Clinic – Mathematics of Medical Imaging and Radiation TherapyFall 2002Weldon A. Lodwick

  2. Abstract This presentation focuses on the methods and application of optimization under uncertainty to radiation therapy planning where it is natural and useful to model the uncertainty of the problem directly. In particular, we present methods for optimization under uncertainty in radiation therapy of tumors and compare their results. Two themes are developed in this study: (1) the modeling of inherent uncertainty of the problems and (2) the application of uncertainty optimization

  3. Objectives of this presentation • To demonstrate that fuzzy mathematical programming (fmp) is useful in solving large, “industrial-strength” problems • To demonstrate the usefulness and tractability of the Jamison & Lodwick and surprise approaches to fmp in solving large problems

  4. OUTLINE • Introduction: The radiation therapy treatment problem (RTP) • Modeling of uncertainty in the RTP • Optimization under uncertainty A. Zimmermann B. Inuiguchi, Tanaka, Ichihashi, Ramik, and others C. Jamison & Lodwick D. Surprise functions IV. Numerical results – A, C and D

  5. I. The Radiation Therapy Problem • The radiation therapy problem (RTP) is to obtain, for a given radiation machine, a set of beam angles and beam intensities at these angles so that the delivered dosage destroys the tumor while sparing surrounding healthy tissue through which radiation must travel to intersect at the tumor.

  6. I. Why Use a Fuzzy Approach? • Boundary between tumor and healthy tissue • Minimum radiation value for tumor a range of values • Maximum values for healthy tissue a range of values • The calculation of delivered dosage at a particular pixxel is derived from a mathematical model • Alignment of the patient at the time of radiation • Position of the tumor at the time of radiation

  7. I. CT Scan - Pixels and Pencils

  8. I. ATTENUATION MATRIX

  9. I. EXAMPLE - Attenuation Matrix • Suppose there are two pencils per beams and two voxels

  10. I. Constraint Inequalities

  11. I. Objective Functions

  12. I. The Fuzzy Optimization Model

  13. II. Modeling of uncertainty in the RTP Four sources of uncertainty and fuzziness in the RTP: • Delineation of tumors and critical tissue • Radiation tolerances or critical dose levels for each tissue type or tumor • Model for the delivered dose, that is the dose transfer matrix • The location of tissue at the time of radiation

  14. II. The RTP process – in practice • The oncologist delineates the tumor and critical structures • A candidate set of beam intensities is obtained; for example by linear programming, fuzzy mathematical programming, simulated annealing, or purely human choice. • These beam intensities are used as inputs to a FDA (Federal Drug Administration) approved dose calculator computer program to produce the graphical depiction of the dose deposition of each pixel (as color scales and dose-volume histograms, DVH’s – see Figure 1).

  15. II. Example Dose Volume Histogram (DVH)

  16. III. Optimization Under Uncertainty The general fuzzy/possibilistic model considered here is:

  17. III. Zimmermann’s approach Translate to a real-value linear program

  18. III. Jamison & Lodwick approach Translate into the nonlinear programming problem

  19. III. Advantages to the J&L approach • If f(x) is convex, then the problem is a convex nlp with simple bound constraints • It optimizes over all alpha-levels; that is, it does not force each constraint to be at the same alpha-level • Large problems can be solved quickly; that is, it is tractable for large problems

  20. III. Surprise function approach Each fuzzy constraint

  21. III. Surprise function approach - continued The fuzzy problem is translated into the nonlinear programming problem This is a convex nlp with simple bound constraints.

  22. III. Why use the surprise function approach? • It is a convex nlp with simple bound constraints • It optimizes over all the alpha-levels; that is, it does not force each constraint to be a the same alpha-level • Large problems can be solved quickly; that is, it is tractable for large problems

  23. IV. Surprise – problem: Black is out of body, blue is critical organ, yellow/green is other critical organs, red is tumor – 32x32 image, 8 angles Set-up time =5.4580 Optimization time= 1.7130

  24. IV. Surprise 32x32 with 8 angles – delivered dosage

  25. IV. Surprise 32x32 with 8 angles - Tumor dvh

  26. IV. Surprise 32x32 with 8 angles – Critical dvh

  27. IV. Surprise 64x64 with 8 angles – delivered dosageSet-up time = 11.0160, optimization time = 2.2930

  28. IV. Surprise 64x64 with 8 angles – Tumor dvh

  29. IV. Surprise 64x64 with 8 angles – Critical dvh

  30. IV. Zimmermann – 32x32 with 8 angles Set-up time = 4.6060 Opt time =171.1060

  31. IV. Zimmermann 32x32 with 8 angles – tumor dvh

  32. IV. Zimmermann 32x32 with 8 angles – critical dvh

  33. IV. Zimmermann – 64x64 with 8 anglesSet-up time = 8.8930, Optimization time =125.1100

  34. IV. Zimmermann: 64x64 with 8 angles – Tumor dvh

  35. IV. Zimmermann: 64x64 with 8 angles – Critical dvh

  36. IV. J & L – 32x32 with 8 angles Setup time - 5.3070Optimization time - 7.3410

  37. IV. J & L 32x32 with 8 angles – tumor dvh

  38. IV. J & L 32x32 with 8 angles – critical dvh

  39. IV. J & L – 64x64 with 8 anglesSet-up time=13.0290, optimization time = 3.145

  40. IV. J & L - 64x64 with 8 anglesTumor dvh

  41. IV. J & L - 64x64 with 8 anglesCritical structure dvh

More Related