Radiotherapy Design

Canadian Cancer Society estimates that nearly 1 out of 3 Canadians will develop cancer in their lifetime. More than half of cancer patients receive radiation therapy as part of their treatment. In external-beam radiation therapy, a patient is typically positioned under a robotic arm of a linear accelerator that delivers radiation; see figure on the left for Elekta radiotherapy treatment unit. High energy radiation has the capacity to destroy cancerous cells; however, the radiation also damages healthy tissues. Therefore, a strategic radiation delivery is required to conform to the shape of the tumour while delivering the minimum amount of radiation to the surrounding healthy organs.

Planning for radiation therapy involves solving an inverse problem. The goal of the inverse problem is to find the best match between a physically-realizable treatment plan and the ideal treatment plan prescribed by a physician. Optimization models and methods are commonly used to accomplish this task.

To an idea how one formulates a mathematical problem whose solution will correspond to a good treatment plan, first consider a simple model for radiation delivery during the treatment. During the planning phase the patient’s body is discretized into a collection of small cubes called voxels; see left figure (a horizontal “slice” of a patient with prostate cancer taken about patient’s pelvis; the figure first depicts part of the CT scan, then the contours of relevant biological structures are superimposed onto a scan, then the CT scan is removed and a planar voxel grid is applied, finally a single voxel is shaded light grey).

The amount of damage caused to either healthy or cancerous cells is proportional to the amount of radiation a cell receives: the higher the dose, the greater the damage. Provided the voxels are small enough (typically 2-3 mm), it may be assumed that within each of the voxel all the cells receive the same amount of radiation. When high-energy photons pass through the patient body, they deposit some amount of energy within each voxel; the precise amount of energy deposited depends on physical characteristics of the tissue. In simple terms, one may think of radiation as light emitted from some source, say, a flashlight, and the patient’s body being made of glass with varying opacity; the darker the glass, the more it will heat up during the prolonged exposure to the light. Now, think of the resulting glass temperature as the amount of radiation energy being absorbed due to the exposure to radiation; see next figure.

In case of radiation therapy, one controls the intensity of each of the radiation beams that go through the patient’s body during the treatment, and thus, the amount of energy that the body will absorb. In the analogy with the opaque glass and the light source, basically, you may choose how long to keep your flashlight on and which angles to choose to shine it from. Fortunately, the law that captures the total energy delivered to a single voxel as a function of radiation beam intensities is quite simple – the cumulative amount of energy follows a simple linear relationship with some given coefficients; see next illustration.

A simplest model for radiotherapy planning may be formulated as a system of linear inequalities, where radiation dose delivered to the tumour has to exceed a certain lethal threshold, while dose to healthy tissues must not exceed some critical threshold value. In this case our goal would be to identify a feasible solution or to prove that no such solution exists. In turn, the above may be formulated as the so-called linear optimization problem, where one asks to minimize some linear functional, say, a total dose to healthy tissues, subject to a set of linear equations and inequalities that describe transformation of radiation beam intensities into dose to each of the voxels and requirements on the dose delivered to the tumour and healthy tissues.

One of the main difficulties associated with optimal treatment planning is the sheer size of the underlying optimization problem --the optimization runs over thousands of beams and hundreds of thousands voxels-- and hence the amount of computations needed to be performed while solving the problem numerically. Furthermore, the search for an optimal solution is complicated by the presence of inherent uncertainties in the model; for example, typically treatment planning is performed based on medical imaging data obtained at least a few hours before the actual treatment, and, therefore, the treatment plan does not account for any physical change in the patient during the treatment itself (such as the displacement of the organs due to breathing and heartbeat), medical imaging data and model parameters include measurement errors, etc.

Most numerically attractive optimization problems are the so-called convex problems, such as minimizing a linear functional over a convex subset of finite-dimensional real vector space. On the other end of the spectra, optimization problems that lack convexity are notoriously difficult (and often even impossible) to solve numerically.

We investigate the use of (convex) optimization models and methods to optimize radiotherapy design with the end goal to allow finding the best possible, i.e., optimal radiotherapy treatment plan for a given patient. Presently within this project we investigate

If you would like to learn more about this project, please contact Dr. Yuriy Zinchenko.

Selected references:

1. Y. Zinchenko, T. Craig, H. Keller, M. Sharpe, T. Terlaky: Controlling the dose distribution with gEUD-type constraints within the convex radiotherapy optimization framework, Physics in Medicine and Biology 53, 1-20, 2008
2. M. Chu, Y. Zinchenko, S. Henderson, M. Sharpe: Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty, Physics in Medicine and Biology 50, 5463-5477, 2005