Group Writing Project: Chemotherapy Dosage
Originally from Math 19, Winter 2002
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
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.
Grading and Resources
This 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.
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