SYLLABUS ‑ MATH 4/5779: Mathematics Clinic

Image Processing and Radiation Therapy

Sponsored by

University of Colorado Health Sciences Center Department of Radiation Oncology

Fall Semester ‑ 2002

Professor: Weldon A. Lodwick

Office: CU-Denver Building, Room 622

Telephone: 556‑8462 (office - voice mail), 556‑8442 (secretary), 556-8550 (fax)

E-Mail: weldon.lodwick@cudenver.edu

Web Site: http://www-math.cudenver.edu/~wlodwick

Office Hours: Tu/Th^* 2:30-3:45 PM Math Education Resource Center (math lab) SI 130

W 9:00-10:00 AM CU-Denver Bldg 622

Other times by appointment

* I may be late once a month (first Tuesday of the month 9/3, 10/1, 11/5 and 12/3) due to faculty assembly meetings. Moreover, I may have to change these times depending on university committee schedules.

Students with Disabilities: If you have a disability that requires accommodation in this course, please see me as soon as possible. I am happy to make appropriate accommodations, provided timely notice is received.

Cell Phones: You are to turn off your cell phones prior to entering class.

Objectives of this Mathematics Clinic – The clinic is primarily a pedagogical tool where one learns applied mathematics by being a part of an applied mathematics effort. Working in research teams to develop results associated with a project (solving a set of problems and presenting the results) is an integral part of every clinic. Thus, we will try to solve problems that are of current concern. In particular, for this semester, our objectives will be:

1. Develop an attenuation matrix generator that can be downloaded from the web for researchers to use to test their algorithms.
2. Develop a problem set of cat scans containing both easy and complicated tumor locations at a variety of resolutions for researchers to test their algorithms.
3. Test a variety of methods for radiation therapy to determine which are the best in obtaining solutions – simulated annealing, tabu search, linear programming, nonlinear programming, quadratic programming, multi-objective programming, variety of objective functions, penalty methods, surprise functions, and optimization under uncertainty methods.
4. Develop a segmentation identifier that has better results than any of those that have previously been developed.
5. Test various algorithms including interior point methods for the radiosurgery problem.
6. Develop materials for a workshop on radiation therapy problems.
7. Develop real time methods for registration of a patient with her/his cat scan.

PROPOSED COURSE OUTLINE

The proposed outline is the initial guess at the topics that will be fruitful to investigate. Research is a process of discovery when one does not know, so the rule is that we will modify our topics during the semester. Thus the proposed outline will undoubtedly change as we learn more during the semester.

Topics:

1. Image Fusion
2. Segmentation – automated and/or semi-automated
3. Radiation Therapy Models
4. Radiosurgery

Problems:

1. Image Fusion: Given two images (cat scans) of the same thing (a patient tumor) taken at two different times,
1. Determine which pixels correspond and compute the probability of false positives.
2. Reconcile a patient’s cat scan taken days before with the patient as s/he is at the time for a radiation therapy treatment (called the registration problem).
2. Segmentation: Given a cat scan, classify all the component parts (organs) of the image.
3. Radiation Therapy Models: Given a tumor, classification of organs, parts and tissues in terms of how sensitive they are to radiation,
1. Web-based Attenuation Matrix Generator and Sample Problems: Develop a web-based attenuation matrix generator along with several cat scan examples at different resolutions (16x16, 32x32, 64x64,…, 512x512) both manually generated (say in paintbrush) and actual that can be used by the class and researchers.
2. Optimization Models: Deliver sufficient radiation to destroy the cancer tumor while delivering no more than the physician specified maximums to the critical organs, parts and tissue while minimizing one or more objectives (minimization of total dosage, maximize minimum radiation to the tumor, minimize tissue complication probabilities, number of beam angles, etc. and combinations of these). That is, determine the path (angles) and intensities for a radiation machine (accelerator) so that the tumor cells are killed and the health tissues are spared. The problem is to get various models up and running and to perform experiments on our sample problems (developed in 3a) making recommendations based on the performance.
3. Fractionated Radiation Therapy Models: Determine how to vary the settings of the accelerator at each of the 20 sessions that are typical of a radiation treatment. This process (of multiple treatment) is called fractionated radiation therapy. So, the problem is how to determine the optimum strategy for fractionation.
4. Radiosurgery: Given a brain growth, deliver many very thin low-intensity beams of radiation to dissolve the growth while sparing all surrounding healthy brain cells. This is the same problem as the radiation therapy problem except that we are looking for a very large number of very low intensity beams. Moreover, the tolerances and exact location of the beams are more critical. For example, over-radiating the medulla oblongata can cause stoppage of breathing or over-radiating of the optic chiasm can cause blindness.
5. Cancer Growth Models: Review the literature and develop a cancer growth model that can be incorporated into the radiation therapy treatment plan. This is especially important for the fractionation problem.

Projects:

The types of projects associated with our clinic are:

1. Development of prototype software.
2. Scour the literature on one or more problem(s) and synthesize the findings with recommendations as to what approaches are most promising.
3. Develop data sets along with data protocols (data structures) that enhance the use of the data.

MY APPROACH TO TEACHING

I believe that teaching is a process that involves an active partnership. My role is that of a guide to your learning. Therefore, I am responsible to open the way, to encourage, and to nudge you toward your own learning. In the context of the math clinic, I will try to model the process of applying mathematics to the radiation therapy problem. I will help guide you toward this learning by providing mathematics for you to experience. It is my aim to communicate mathematics in a way that is supportive and nurturing of your efforts. Your role is to find a way to experience and articulate the mathematics that is presented and that you encounter. I believe that it is your responsibility to let me know when you find yourself not understanding mathematical concepts that are presented in class. Once you make this known, it is our responsibility to work on trying to attain clarity. I will try to be as proactive as possible. I believe that results on projects give us the opportunity to clearly see where the areas of mathematical understanding are and what areas need more attention.

OUTCOMES

By the end of the semester you should be able to read, understand and apply appropriate methods associated with aspects of medical image processing and radiation therapy treatment planning we’ve studied this semester (image fusion, segmentation, radiation therapy, and cancer growth models) to correctly solve associated problems. Secondly, given a problem medical image processing or the radiation therapy problem, you should be able to: (i) translate the description of the problem into an algorithm, (ii) choose and apply the appropriate software method(s), (iii) obtain the correct solution(s), and (iv) (correctly) interpret and display results. Lastly, by the end of the semester you should be able to judge, for yourself, the veracity of statements made in the areas of our study.

EVALUATION

Each person on a team will execute a project (identify a set of problems, find solution methods, present the results and write-up the results). In particular, the following are components that will be evaluated.

1. Participation – attendance and contributing, intermediate results (20%)
2. Annotated bibliography (5%)
3. Final presentation (15%)
4. Software development (results) (20%)
5. Testing and analysis (20%)
6. Written report (20%)
1. Presentation slides (power-point for example)
2. Chapter in the clinic report

** Graduate students will have extended content and be held to higher standards.

The grade assignments are on the 10 percent scale (A = 90%-100%, B = 80%-89%, C = 70%-79%, D = 60-69%).

IMPORTANT DATES:

Team selection – on or before August 29^th

Project selection and proposal - on or before September 5^th

Division of labor - on or before September 6^th

First quarter reports – September 17^th and 19^th

Annotated bibliography 1 – September 20^th

Testing and analysis 1 – October 4^th

Midterm reports – October 8^th and 10^th

Third quarter reports - November 12^th and 14^th

Annotated bibliography 2 – November 22^nd

Testing and analysis 2 – November 29^th

Presentation – December 3^rd and 5^th

Final reports – December 9^th

General advice: Keep all materials that I turn back in case you think I have not credited you with the points you earned. I can only correct your score if you have what I have turned back to you. It is a good idea to xerox anything that you turn in just in case I lose what you turn in. Please check to make sure that the points you earned are the points I have recorded. Note: The statistics that I have read about correctness of professors in recording grades state that there is a 6% error rate in our recording of your grades. Please make sure that I have correctly recorded your points.

POLICIES

Drops and incomplete grades: See Schedule of Courses for the relevant dates with respect to dropping this course. The incomplete policy of the Mathematics Department and the College of Liberal Arts and Sciences is strictly enforced. Incomplete grades are given only in situations in which a student who has been in good standing all semester, is prevented from completing a course assignment (for example the final exam) due to circumstances beyond her/his control (for example, hospitalization, jury duty, revised job assignments, death in the family).

Legitimate Excuses: Legitimate excuses for missing tests and quizzes are for reasons that are beyond your control. You may be required to produce an official, signed excuse. If you are needed in a wedding, for example, you must talk to me prior to the (blessed) event. If you are legally arrested, then this is not a legitimate excuse. For matters that are within your control, the general rule is that it is not excused. However, talk to me prior to the event.

INSTRUCTIONS FOR PROJECTS (will come later after we decide what to do)

A project consists of:

1. Proposal – Each group must turn in a proposal to me (this can be done via email) outlining what the group wishes to do, the scope of the work and the outcomes to be delivered. This is like writing specs for a project on which you are bidding.

2. Division of labor – this is a set of tasks and subtasks that is assigned to individuals in your group with an associated due-date. It is a formal contract between the members of the group. Once you have your proposal approved and you have written your division of labor, the group needs to schedule of meeting with me so that we can go over the division of labor, its associated responsibilities and expectations.

Software development (or research – there will be a different breakdown for research, please talk to me if you are interested in doing research) consists of code, input, execution, output, ease-of-use, and documentation.

1. Code - the actual computer implementation of the project. Attention must be paid to efficiency, readability and portability.
2. Input – the way information is passed to the software must be compelling to the client.
3. Execution - the algorithm as run must correctly perform what it was designed to do.
4. Output - relevant, clear display of solution (tables, graphs, images).
5. Ease – the software must be easy to use.
6. Documentation – a hardcopy of the documentation on how to use the software. Moreover, help files must be part of the software.

3. Report that will be a chapter in the Clinic Report (instructions to be given later)

4. Testing and analysis consists of two parts – testing and analysis.

a. Testing - this part simply consists of running the software developed on the test problems

b. Analysis - the purpose of an analysis is to get you to critically evaluate the results obtained from the software as it was run on the test problems. Part of an analysis is a critique of the software.

Working in Groups for Mathematics students

Your grade will depend, for a large part, on your success as a member of a group. When Mathematicians work in the “real world” they often have to work in groups such as this. Groups can be innovative, the unique combination of skills and experiences may lead to new insights, new ideas. In addition, you can accomplish far more than one person working on his or her own. But working in groups requires skills -- social skills and skills of understanding yourself. Take this opportunity to learn, not only Mathematics, but also about yourself and your classmates.

Below are some guidelines and suggestions to help you get your groups off to a good start. Student groups can be productive and creative, or they can be frustrating and end up in conflict. If you pay attention to how the group works, and the needs of the members, you may find that the group works more effectively. An effective group does not happen by accident. You can contribute to helping your group success with its project by taking a few minutes each meeting to take care of group process needs. Just as you take care of your car, to make sure it has fuel, that its oil is changed, you need to take care of your group process.

All meetings:

Make sure everyone has been invited, and that it is clear where you will meet. A group member may set up the agenda, or the group may set it at the end of the last meeting. The agenda must include a review of the previous meeting, and a statement of the purpose of the current meeting. Someone needs to be responsible for taking notes, and someone can help keep the group on track.

There will be disagreements in your group. It is natural that people will see different perspectives. Take the disagreements as a sign that people have different ideas, backgrounds and knowledge, and give time to the discussions that are important to the task. Give people time to think about different perspectives. Do not force rapid decisions. Take advantage of e-mail and other technologies (telephone) to thoroughly air out viewpoints.

First meeting:

Take a few minutes to introduce yourselves, to discuss what your interests are, both inside and outside of class. Talk about where you are from, what brought you to UCD, and talk about how you prefer to work, what roles you prefer to take, what skills you bring. This is not bragging, but more of an assessment of what tools and skills the group has. If people have strengths and preferences, it makes sense that the whole group will benefit from those skills. (This is very important, do not skip it for any reason.) The group may continue this discussion through e-mail; to be sure that everyone has a chance to hear everyone else.

Decide how you want to meet - do you want to have a leader? Maybe you can take turns leading, or choose one person who is responsible for making sure everyone participates, that everyone is clear on their roles, and that any problems that come up are addressed by the group. Choose a note-taker, and exchange phone numbers and e-mail addresses. Agree on a time and place to meet. Then continue with the task part of the meeting.

Second meeting

Review what you did during the first meeting, and any activity that occurred between the meetings. Review the purpose of the group. Make sure that everyone is clear and agrees on the purpose. For people in the group who prefer it, make sure that e-mail channels are available for discussion. Set ground rules for the group. This does not need to take too much time, but agree on basic guidelines such as listening to one another, focusing on the group goal, and making decisions by consensus.

It is important that all individuals in the group feel included. This is difficult at times, when there is pressure to complete a task, and if individual or cultural styles are different. Often times, a few people will band together, and exclude the others, who may work differently. Differences occur in the way people use time, their openness to communication, and even in approaches to problem solving. However, if the group decides that it is important to include everyone, and works on building an identity and process that is inclusive, those differences can lead to new ideas and creative approaches for solving problems.

The group may wish to give itself a name. This may seem to be a distraction, but is actually a good way for the group to work together for practice. Then continue with the task part of the meeting.

Third meeting

Review what has occurred in the last meetings, and review the group goal. Assess where the group is now, and how far it has to go. If it is appropriate, you may wish to create a timeline (if you haven’t already). Check with every member to ensure that each person is clear on his or her part of the task. Check to be sure that everyone has been heard from, either in the meetings, or through e-mail. If there are group problems, be sure that they are addressed in a non-judgmental way. If something goes wrong, if some assignment is not done, do not blame, but find out what the blockage was. Then proceed with the tasks.

Subsequent meetings

Review previous meetings and communications

(Discuss problems and issues as problems to be solved)

Reports from each member

Discuss what has to happen next

Make assignments

Set next meeting agenda