A Tutorial on Radiation Oncology and Optimization

Designing radiotherapy treatments is a complicated and important task that affects patient care, and modern delivery systems enable a physician more flexibility than can be considered. Consequently, treatment design is increasingly automated by techniques of optimization, and many of the advances in the design process are accomplished by a collaboration among medical physicists, radiation oncologists, and experts in optimization. This tutorial is meant to aid those with a background in optimization in learning about treatment design. Besides discussing several optimization models, we include a clinical perspective so that readers understand the clinical issues that are often ignored in the optimization literature. Moreover, we discuss many new challenges so that new researchers can quickly begin to work on meaningful problems

Optimal Treatments for Photodynamic Therapy

Photodynamic therapy is a complex treatment for neoplastic diseases that uses the light-harvesting properties of a photosensitizer. The treatment depends on the amount of photosensitizer in the tissue and on the amount of light that is focused on the targeted area. We use a pharmacokinetic model to represent a photosensitizer\u27s movement through the anatomy and design treatments with a linear program. This technique allows us to investigate how a treatment\u27s success varies over time

Fast and Robust Techniques for the Euclidean p-Median Problem with Uniform Weights

We discuss new solution techniques for the p-median problem, with the goal being to improve the solution time and quality of current techniques. In particular, we hybridize the discrete Lloyd algorithm and the vertex substitution heuristic. We also compare three starting point techniques and present a new solution method that provides consistently good results when appropriately initialized

Simultaneous Data Perturbations and Analytic Center Convergence

The central path is an infinitely smooth parameterization of the non-negative real line, and its convergence properties have been investigated since the middle 1980s. However, the central path followed by an infeasible-interior-point method relies on three parameters instead of one, and is hence a surface instead of a path. The additional parameters are included to allow for simultaneous perturbations in the cost and righ-hand side vectors. This paper provides a detailed analysis of the perturbed central path that is followed by infeasible-interior-point methods, and we characterize when such a path converges. We develop a set (Hausdorff) convergence property and show that the central paths impose an equivalence relation on the set of admissible cost vectors. We conclude with a technique to test for convergence under arbitrary, simultaneous data perturbations

Navy Personnel Planning and the Optimal Partition

One could argue that the Navy\u27s most important resource is its personnel, and as such, workforce planning is a crucial task. We investigate a new model and solution technique that is designed to aid in optimizing the process of assigning sailors to jobs. This procedure attempts to achieve an increased level of sailor satisfaction by providing a list of possible jobs from which a sailor may choose. We show that the optimal partition provided by an interior-point algorithm is particularly useful when designing the job lists. This follows because a strictly complementary solution to the linear programming relaxation observes all possible optimal solutions to the original binary problem. The techniques developed rely on a continuous parametric analysis, and we show that the parameterization provides meaningful information about the structure of the optimal assignments

Partitioning Multiple Objective Optimal Solutions with Applications in Radiotherapy Design

The optimal partition for linear programming is induced by any strictly complementary solution, and this partition is important because it characterizes the optimal set. However, constructing a strictly complementary solution in the presence of degeneracy was not practical until interior point algorithms became viable alternatives to the simplex algorithm. We develop analogs of the optimal partition for linear programming in the case of multiple objectives and show that these new partitions provide insight into the optimal set (both pareto optimality and lexicographic ordering are considered). Techniques to produce these optimal partitions are provided, and examples from the design of radiotherapy plans show that these new partitions are useful

Radiotherapy Treatment Design and Linear Programming

Intensity modulated radiotherapy treatment (IMRT) design is the process of choosing how beams of radiation will travel through a cancer patient to treat the disease, and although optimization techniques have been suggested since the 1960s, they are still not widely used. Instead, the vast majority of treatment plans are designed by clinicians through trial-and-error. Modern treatment facilities have the technology to treat patients with extremely complicated plans, and designing plans that take full advantage of the technology is tedious. The increased technology found in modern treatment facilities makes the use of optimization paramount in the design of successful treatment plans. The goals of this work are to 1) present a concise description of the linear models that are under current investigation, 2) develop the analysis certificates that these models allow, and 3) foreshadow future research avenues

Designing Radiotherapy Plans with Elastic Constraints and Interior Point Methods

A new linear programming model used to aid in the design of radiotherapy plans is introduced. This model incorporates elastic constraints, and when solved with a path following interior point method, produces favorable plans. A sound mathematical analysis shows how to interpret the solution, and hence, the treatment planner receives meaningful knowledge about the radiotherapy plan being developed. Preliminary experiments are conducted

An Introduction to Systems Biology for Mathematical Programmers

Many recent advances in biology, medicine and health care are due to computational efforts that rely on new mathematical results. These mathematical tools lie in discrete mathematics, statistics & probability, and optimization, and when combined with savvy computational tools and an understanding of cellular biology they are capable of remarkable results. One of the most significant areas of growth is in the field of systems biology, where we are using detailed biological information to construct models that describe larger entities. This chapter is designed to be an introduction to systems biology for individuals in Operations Research (OR) and mathematical programming who already know the supporting mathematics but are unaware of current research in this field

The Asymptotic Optimal Partition and Extensions of the Nonsubstitution Theorem

The data describing an asymptotic linear program rely on a single parameter, usually referred to as time, and unlike parametric linear programming, asymptotic linear programming is concerned with the steady state behavior as time increases to infinity. The fundamental result of this work shows that the optimal partition for an asymptotic linear program attains a steady state for a large class of functions. Consequently, this allows us to define an asymptotic center solution. We show that this solution inherits the analytic properties of the functions used to describe the feasible region. Moreover, our results allow significant extensions of an economics result known as the Nonsubstitution Theorem