1 The Use of Fuzzy Optimization Methods for Radiation Therapy of Tumors Weldon A. Lodwick

2 Abstract This presentation focuses on the application of fuzzy optimization algorithms to large optimization problems where it is natural and useful to model the uncertainty of the problem directly. In particular, we apply fuzzy optimization to radiation therapy of tumors. Two themes are developed in this study: (1) the modeling of inherently fuzzy problems directly as fuzzy optimization problems and (2) the application of fuzzy optimization to radiation therapy of tumors.

3 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

4 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 voxel is derived from a mathematical model Alignment of the patient at the time of radiation Position of the tumor at the time of radiation

5 CT Scan - Voxels and Pencils

6 EXAMPLE - Attenuation Matrix Suppose there are two pencils per beams and two voxels

7 Constraint Inequalities

8 Fuzzy Tumor Location Constraint

9 Objective Function

10 The Fuzzy Optimization Model

11 Simple Fuzzy Model

12 Transformation to a Fuzzy Linear Programming Problem

13 General Fuzzy Model

14 Alpha-cuts

15 Expected Average

16 The Transformed Problem

17 Method The transformed problem is a nonlinear programming problem Any descent method will work. The maximum with respect to alpha cuts can be computed apriori.

18 Application Applied crisp versions to day-to-day radiation therapy Computerized Medical Systems (Saint Louis) implements the crisp version, out in June, 1999 Fuzzy version experiments are beginning We are developing MATLAB plug-in attenuation matrices - the A matrix