## Group Writing Project: Chemotherapy Dosage

### The Problem

You 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:

T(t) = 1.1^(1-(3 D(t)/T(t-1))) T(t-1)
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 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 paper

In 3-5 pages, your group should explain the problem that you are solving and your solution to it. Your paper should have three parts:
• Introduction: describe the problem. You don't need to rehash the medical background that I have put above, since your audience (myself and the doctors) already know it. But you should explain the model. So describe what each variable represents and the initial conditions. Explain the equations that relate the variables. What would happen to the tumor size and the immune system without any chemotherapy? How long would the patient live without treatment? When the patient receives a dose of chemotherapy, how does this affect T(t) and I(t)? What can you say about their rates of change?

• Main body: describe your recommendations. Explain how your analysis of the equations led you to recommend a dosing schedule. If you did any calculations aside from what the worksheet does, include them and explain their meaning. Include a table of values for D(t) going from week 1 until week 52. Include graphs (which can be drawn directly from your Excel worksheet) of T(t) and I(t) during that time. But in addition to these numerical and graphical descriptions, you should also describe your recommended dosing schedule in words. What is the idea behind this treatment? Why is it the best thing that can be done for the patient? If the cancer is not yet cured after 52 weeks, how should the doctors proceed after that? What will be the limits of T(t) and I(t) as t approaches infinity?

• Conclusion. Sum up your solution to the problem, and generalize it. How should the doctors choose the right dosages for patients with bigger tumors, or with weaker immune systems? Will the same approach still work? For extra credit, you can try to figure out the largest initial tumor size T(0) that can be cured when I(0)=9.