Suppose you work for a drug company that has recently discovered a new cancer drug. It has been tried out on a small number of patients and now you need to study the results further before proceeding to widespread clinical trials. The model describing the relative (as a percentage of maximal possible tumor size, thus dimensionless) radius \(r\) of a tumor for a dosage of relative size \(k\geq 0\) is given by the differential equation
where \(\alpha\geq 0 \) is a parameter called the intrinsic growth rate. The time units are months. Suppose, throughout this worksheet, that at time of detection \(r=.05\text{.}\)
1.
Explain why the expressions \(\alpha\dfrac{(1-r)r}{1+r}\) and \(kr\) are both always positive. Develop an Insight with one inflow, one outflow, and two variables (with sliders). (For students turning this in, you will need to submit your Insightmaker files.)
2.
Use the model you have developed above to describe the long-run behavior of the tumor under each of the following scenarios:
\(k=0\) (no treatment)
\(\alpha = 0\) (no growth)
Explain why each of these results makes practical sense.
3.
Determine the equilibrium solutions of the given model algebraically (show your work) and determine their stability depending on the values of \(\alpha\) and \(k\text{.}\) What is the desirable (in terms of real-world cancer treatment) in terms of the equilibria and stability of them, and how must \(\alpha\) and \(k\) be related to achieve this outcome. Include graphs from Insightmaker simulations to illustrate each possible outcome with descriptions that are accessible to a non-mathematical audience.
4.
Suppose that after detection Patient #1 was given a regular dose of the drug with \(k=1.5\) and their tumor size grew according to the following table:
Table2.3.1.Tumor Size for Patient #1
Time (months)
\(r\)
1
0.091677
2
0.152964
3
0.196308
4
0.217980
5
0.227098
6
0.230665
Use a converter for this data and the method of least-squares (recall how this was done in Section 2.1) to determine the intrinsic growth rate for this tumor.
5.
(Ethical Question) Suppose that possibly deadly side effects occur when the drug is administered with \(k\gt 2.5\text{.}\) Based on the results of patient #1, should widespread clinical trials proceed? Try to describe several considerations in your answer from the point of view of multiple stakeholders.
6.
How realistic does this model feel? What aspects seem realistic and which ones don’t? You can answer this mathematically as well as by doing a little internet research about cancer treatment.
