# Case Study: Optimizing Intensity-Modulated Radiation Therapy Treatment Planning Problem

** Case study background and problem formulations**

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

**PROBLEM: problem_men**

Minimizing Pm2_pen_g (minimizing partial moment two penalty function)

subject to

Box constraints (bounds on variables)

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Pm2_pen_g = Partial Moment Two Penalty for Gain

Box constraints = constraints on individual decision variables

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Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 10 | 10 | 0.00035009773 | <0.01 |

Dataset2 | Problem Statement | Data | Solution | 1,113 | 20,859 | 15.490633332895 | 312.76 |

Dataset3 | Problem Statement | Data | Solution | 2,736 | 44,362 | 5.379941523030 | 879.34 |

Dataset4 | Problem Statement | Data | Solution | 2,055 | 37,794 | 3.875103565617 | 103.19 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 10 | 10 | 0.000350098 | <0.01 |

Dataset2 | Matlab code | Data | Solution | 1,113 | 20,859 | 15.4906 | 306.52 |

Dataset3 | Matlab code | Data | Solution | 2,736 | 44,362 | 5.37994 | 864.84 |

Dataset4 | Matlab code | Data | Solution | 2,055 | 37,794 | 3.87768 | 91.27 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 10 | 10 | 0.000350098 | <0.01 |

**CASE STUDY SUMMARY**

This case study solves an intensity-modulated radiation therapy (IMRT) treatment planning problem. The problem statement is presented in the paper “An exact approach to direct aperture optimization in IMRT treatment planning” by Men et al., (2007). In the original case study, the sum of squares of penalties is minimized to reduce the radiation therapy damage. PSG uses the “partial moment two penalty” (pm2_pen) function, which is the average of sum of squares of penalties. Therefore, the optimal point obtained with PSG is the same as in the original case study; however, the PSG objective value differs from the original objective value by a fixed multiplier.

The scenarios matrices in radiation therapy case studies are sparse (few non-zero elements). Therefore, packed matrix format (pmatrix) is quite beneficial for these problems. We solve four problems which differ only by the dataset (all problems have the same mathematical formulation). Dataset1 has no connections with real life problems: it is demonstrative, the rest three dataset are of the big size and represent real life problems.

Therefore problem is solved using 4 datasets:

• Dataset1 for “short case study” including 10 variables and matrix of scenarios with 10 scenarios;

• Dataset2 including 1,113 variables and matrix of scenarios with 20,859 scenarios;

• Dataset3 including 2,736 variables and matrix of scenarios with 44,362 scenarios;

• Dataset4 including 2,055 variables and matrix of scenarios with 37,794 scenarios.

• Dataset1 for “short case study” including 10 variables and matrix of scenarios with 10 scenarios;

• Dataset2 including 1,113 variables and matrix of scenarios with 20,859 scenarios;

• Dataset3 including 2,736 variables and matrix of scenarios with 44,362 scenarios;

• Dataset4 including 2,055 variables and matrix of scenarios with 37,794 scenarios.

**References**

• Men, C., Romeijn, E., Taskin, C. and J. Dempsey, (2007): An Exact Approach to Direct Aperture Optimization in IMRT Treatment Planning, in Physics in Medicine and Biology, Vol. 52 , pp. 7333–7352.