8.3 Radiation Therapy: An Application of Linear Optimization

Course Info

Instructor

Prof. Dimitris Bertsimas

Departments

Sloan School of Management

As Taught In

Spring 2017

Level

Graduate

Topics

Learning Resource Types

Lecture Videos

Lecture Notes

Problem Sets with Solutions

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8 Linear Optimization

Video 1: Introduction

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The slides from all videos in this lecture can be downloaded here: An Application of Linear Optimization (PDF - 1.8MB).

The images in this video of the head scans and images and data used in other videos in this lecture, come from the Computational Environment for Radiotherapy Research (CERR), Copyright 2010, Joseph O. Deasy, on behalf of the CERR development team. CERR development has been led by: Aditya Apte, Divya Khullar, James Alaly, and Joseph O. Deasy. CERR has been financially supported by the US National Institutes of Health under multiple grants, and is distributed under the terms of the Lesser GNU Public License.

Quick Question

In the next video, we’ll be formulating the IMRT problem as a linear optimization problem. What do you think the decision variables in our problem will be?

The amount of radiation to deliver to the tumor close

The intensities of the beams close

The intensities of the beamlets check

The shape of the tumor close

Explanation We get to decide the beamlet intensities - these will be the decision variables in our optimization problem. The amount of radiation to the tumor will be computed using the beamlet intensities, but we also want to make sure we know the amount of radiation to healthy tissue. The intensities of the beams would have been the decision variables in traditional radiation therapy, and the shape of the tumor is data.

Continue: Video 2: An Optimization Problem

Quick Question

In the previous video, we constructed the optimization problem (see the last slide).

If the beamlet intensity of the first beamlet is set to 3, how much radiation will that beamlet deliver to tumor voxels?

Exercise 1

Numerical Response

How much radiation will it deliver to healthy tissue voxels?

Exercise 2

Numerical Response

Explanation

Beamlet 1 hits one tumor voxel, and two healthy tissue voxels. At unit intensity, it delivers a dose of 2 to the tumor voxel, a dose of 2 to the first healthy tissue voxel, and a dose of 1 to the second healthy tissue voxel. At intensity 3, this means that it will deliver a dose of 2*3 = 6 to the tumor voxel, and 2*3 + 1*3 = 9 to the healthy tissue voxels.

CheckShow Answer

Quick Question

In our optimal solution, we are giving the maximum allowed dose to the spinal cord (5). If we were to relax this, how much could we decrease the objective? Change the right-hand-side (RHS) of the spinal cord constraint to 6, and re-solve the model. By how much did we decrease the objective? (Hint: the previous objective value was 22.75)

Exercise 1

Numerical Response

Explanation

If you change the RHS of the spinal cord constraint to 6 and re-solve the model (Tools->Solver, then hit solve) the new objective is 22.1666667. So we decreased the objective by 0.58333333.

CheckShow Answer

Quick Question

In the previous video, we discussed a Head and Neck case with 132,878 voxels total, 9,777 voxels in the tumor, and 328 beamlets.

How many decision variables does our optimization model have?

Exercise 1

Numerical Response

Explanation

Our decision variables are for the intensities of the beamlets. So in this case, we would have 328 decision variables, which is the number of beamlets.

CheckShow Answer

Quick Question

In your spreadsheet from Video 3, make sure that you have solved the original small example problem (change the spinal cord limit back to 5 and re-solve if you have changed it, and make sure the objective value is 22.75).

Now, change the weight for the spinal cord term in the objective to 5.

Without re-solving, what does the objective value of the current solution change to?

Exercise 1

Numerical Response

Explanation

The term SUMPRODUCT(B14:B19;F5:F10) in the objective (corresponding to Voxel 5) should now be 5*SUMPRODUCT(B14:B19;F5:F10). This changes the objective value to 42.75.

Now re-solve the model. What does the objective change to?

Exercise 2

Numerical Response

Explanation

You can resolve the model by going to Solver, and just hitting Solve. The new optimal objective function value is 25.666667.

Notice how we are now giving a smaller dose to the spinal cord!

CheckShow Answer

Video 2: An Optimization Problem

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Check That You Are Minimizing

If your optimal objective function value was 35.666, then you probably are maximizing instead of minimizing! Be sure to check that you are minimizing the objective in the Solver window.

The images in this video of the brain scans come from the Computational Environment for Radiotherapy Research (CERR).

Video 3: Solving the Problem

In this video, we’ll be using the spreadsheet IMRT_SimpleExample. If you are using LibreOffice or OpenOffice, please download and open the spreadsheet IMRT_SimpleExample (ODS). If you are using Microsoft Excel, please download and open the spreadsheet IMRT_SimpleExample (XLSX). The following spreadsheets have the completed model as it is at the end of the video: IMRT_SimpleExample_Complete (ODS) and IMRT_SimpleExample_Complete (XLSX).