# Optimization Models for the Beam Intensity Optimization Problem

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## Optimization Problems in IMRT

In IMRT planning, a set of parameters, such as number of beams, orientation of beams, intensity of each beam etc. need to be decided for the delivery of a certain radiation dose to the patient. This is done by treatment planner with the aid of an inverse planning system with an optimization engine. Some parameters are chosen by the planner while the others are calculated by the inverse planning system. The computer optimization can potentially simplify the tedious planning procedure and yield the best possible plan. The decision of the parameters is of crucial importance for the quality of the radiation plan.

Outline of the Thesis

Optimization in IMRT is too wide to address completely. In this thesis, we focus on the beam intensity optimization problem, i.e., given the number of beams and the direction of each beam, we optimize the intensity of each beam. Since the goals of delivering a sufficiently high uniform dose to the tumor and a small dose to the OARs and normal tissue contradict each other, we cannot achieve them at the same time. This makes the beam intensity optimization problem inherently multicriterial, i.e., there is no single best solution, instead, there are many best compromises. Therefore, we formulate the beam intensity optimization problem as a multiobjective linear programming model and propose solution methods to solve big multiobjective linear programmes (MOLPs).

### Linear and Nonlinear Programming Models

The first linear optimization model that was developed to aid radiation therapy design appeared in the literature in 1968 (Bahr et al. (1968)). Since then, many researchers have experimented with linear models. See, e.g, Hodes (1974), Langer (1987), Powlis et al. (1989), Legras et al. (1982), Morrill et al. (1990b), Morrill et al. (1991a) and Rosen et al. (1991). For an overview of these models, the reader is referred to Shepard et al. (1999). Basically, all objectives of linear programming models are variations of: (1) minimize average/maximum dose or deviation from upper bounds on dose to critical organs and normal tissue, (2) maximize average/minimum dose to tumor, and (3) minimize average/maximum deviation from prescribed dose to target.

Dose Volume Constraints

Many researchers work on the incorporation of dose volume constraints (DVCs) into the mathematical models. Using linear programming to include DVCs was started by Langer (1987), who was the first to incorporate DVCs in the literature. The author produces a sequence of LPs by enumerating all the combinations of constraint points that satisfy dose-volume constraints, solves these LPs and chooses the voxel combination that gives the best objective value. It is difficult to solve the problem if there are more than 20 voxels. For example, consider the dose-volume constraint of “no more than 40% volume of the lung can exceed a radiation dose of 20 Gy”. Then, if there are 100 voxels in the lung, in order to satisfy the above dose volume constraint we need to solve 100 40 LPs.

#### Mixed Integer Programming

By introducing binary variables into the model it is straight forward to impose dose-volume constraints. By setting these binary variables to 0 or 1, it is possible to enumerate the number of voxels that receive a dose lower than a threshold or higher than a threshold in a specified structure, thus it is possible to model dose-volume constraints. These additional variables turn a linear programme into a mixed integer programme (MIP). Here, we consider an example to show how to convert a dose constraint into a dose-volume constraint. In linear programming, a single upper bound on the critical organ may be imposed as follows: a (i) x CUBi ∀i ∈ C. (2.16) Sometimes, such a constraint can impose infeasibility.

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Multiple Objective Programming

In radiotherapy, the desired dose distribution can not always be obtained, due to physical limitations and to the existence of trade-offs between the various conflicting treatment goals. This multiobjective character of inverse planning has been recognized in radiation therapy optimization only in the last 10 years even though radiotherapy has been applied for several decades. One way to solve a multiple objective (MO) problem is by transforming it into a single objective problem using a specific set of weighting factors for each objective. This is the most prevalent method and is called a priori method for solving MO optimization problems because preference information is incorporated into the model in the form of weights prior to optimization.

Contents :

• Abstract
• Acknowledgements
• Table of Abbreviations
• 1 Introduction
• 1.1 The Treatment Process of IMRT
• 1.2 Treatment Planning
• 1.3 Optimization Problems in IMRT
• 1.4 Outline of the Thesis
• 2 Optimization Models for the Beam Intensity Optimization Problem
• 2.1 Introduction
• 2.2 Dose Calculation and Prescription
• 2.3 Feasibility Problem and Algorithms in IMRT
• 2.4 Linear and Nonlinear Programming Models
• 2.5 Dose Volume Constraints
• 2.6 Mixed Integer Programming
• 2.7 Multiple Objective Programming
• 2.8 Summary
• 3 An MOLP Model for the Beam Intensity Optimization Problem and Solving an MOLP in Objective Space
• 3.1 Notation
• 3.2 An MOLP Model for the Beam Intensity Optimization Problem
• 3.3 Introduction to Multiple Objective Linear Programming
• 3.4 Benson’s Outer Approximation Algorithm
• 3.5 Improvements to Benson’s Algorithm
• 3.6 Summary
• 4 Approximately Solving Multiobjective Linear Programmes in Objective Space
• 4.1 Approximation Version of Benson’s Algorithm
• 4.2 Numerical Results
• 4.3 Summary
• 5 A Dual Variant of Benson’s Outer Approximation Algorithm
• 5.1 Introduction
• 5.2 Geometric Duality
• 5.3 Extension of Benson’s Outer Approximation Algorithm
• 5.4 The Dual Variant of Benson’s Algorithm
• 5.5 Weight Set Decomposition
• 5.6 Numerical Results
• 5.7 Summary
• 6 Approximating the Nondominated Set of an MOLP by Approximately Solving its Dual Problem
• 6.1 Introduction
• 6.2 Further Analysis of the Dual Variant of Benson’s Algorithm
• 6.3 Obtaining the Nondominated Facets of P from D
• 6.4 Solving the Dual MOLP Approximately
• 6.5 Numerical Results
• 6.6 Summary
• 7 Finding Representative Nondominated Points in Multiobjective Linear Programming
• 7.1 Introduction
• 7.2 Quality of Discrete Representations
• 7.3 Existing Methods
• 7.4 Revised Normal Boundary Intersection
• 7.5 Numerical Results
• 7.6 Summary
• 8 Case Study
• 8.1 Reducing the Computation Time
• 8.2 3D Clinical Cases
• 8.3 Decision Support
• 8.4 Summary
• 9 Conclusion

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Multiple Objective Linear Programming in Radiotherapy Treatment Planning