Group Writing Project: Chemotherapy Dosage
Originally from Math 19, Winter 2002
The ProblemYou and your group were just hired by the Stanford Medical Center to consult them about the treatment of Hodgkin's disease -- a form of cancer that affects the lymph nodes and spleen. In this form of cancer, tumors grow in many places at once, which makes it impractical to remove them surgically. Instead, Hodgkin's disease is usually treated with chemotherapy. Chemotherapy isn't very accurate: it kills all cells that are rapidly dividing; most of these cells are cancerous but many are not. So it has serious side effects: nausea, vomiting, and fever among them. Worst of all, it inhibits the body's ability to make red and white blood cells, which damages the immune system. (Many thanks are due to Dr. Maria Seliverstov for explaining the medical information in this project.)
When Hodgkin's disease is diagnosed early, it can be defeated by relatively small doses of chemotherapy that don't threaten the patient's life. In this case, well over 90% of people are living cancer-free after 5 years. But when the cancer is only caught at a late stage, the tumors are too large and numerous to be choked off by small doses of chemotherapy; any dose that is large enough to be effective will also have life-threatening side effects. It is in this case that the Stanford Medical Center needs your help.
Doctors at the Medical Center have come up with a fairly simple model that describes the effect of chemotherapy on tumor size and the immune system. At time t (measured in weeks), the total size of the tumors is T(t) and the strength of the immune system is I(t). The dose of chemotherapy that a patient receives during any given week is D(t). The dose D(t) affects the way the tumors grow or shrink and the way the immune system strengthens or weakens. Thus the equations involve T(t-1) and I(t-1), the values of those variables from the week before:
I(t) = 3 + 0.7 I(t-1) - D(t)
A perfectly healthy person will have no tumor and an immune strength of 10. A patient will die if the tumor size gets bigger than 20, and he or she will be very likely to pick up some deadly infection if I(t) drops below 1. You need to help the Medical Center treat Hodgkin's disease in a patient whose initial tumor size is 8 and whose immune system is 90% healthy -- that is, T(0) = 8 and I(0)= 9.
Your jobYour job is to recommend a dosing schedule: to tell the Medical Center how much chemotherapy the patient should receive each week. So you need to figure out the dosage that will shrink the tumor without killing the patient's immune system, and that will do this as quickly as possible. At a very minimum, your dosage should keep the patient alive for a year (52 weeks), but it would be much better to cure the tumors completely in this time. This is possible, given the numbers above.
To simplify your computations, I will e-mail you a Microsoft Excel worksheet that has all the equations programmed in. All you need to do is enter in the dose of chemotherapy D(t) for each week, and it will calculate the rest. (The dose can be any non-negative real number.) But this doesn't make the problem trivial -- your group will still have to analyze the equations and be creative to find a cure!
When you have decided what dosing schedule to recommend, your group needs to write a 3-5 page paper explaining and justifying your recommendations. After all, the point of this project is not only to develop your problem-solving skills but also to give you practice at communicating technical information in a clear and understandable way.
The paperIn 3-5 pages, your group should explain the problem that you are solving and your solution to it. Your paper should have three parts:
Grading and ResourcesThis project is worth 100 points, out of 500 for the course. Half your grade will be based on the mathematical accuracy of what you write: the answers to the questions I posed above, as well as the effectiveness of the treatment you recommend. The other half of the grade will be for writing quality. I believe it's very important that your writing be clear and readable. If you're not sure exactly how to write a mathematical paper, I would like to point you to an excellent guide available on the Web:
You are also welcome to come consult with me or Jessica. I would be happy to read and comment on paper drafts if you get them to me by Monday, March 4th. The papers are due on Friday, March 8th.
dfuter at math stanford edu Last modified: Mon Dec 6 17:44:14 PST 2004