Drugs used to treat cancer in calculus

Question

A drug used to treat cancer is effective at low doses, with efficacy that increases with the quantity of the drug. However, at sufficiently high doses, the drug becomes lethal. For positive values of the constants k1 and k2, the fraction of patients surviving cancer with this drug treatment is given by

k1 + q k2 + q

where q is the drug quantity in milligrams per day given to the patient.

1. (a) Which is larger, k1 or k2? Why?
2. (b) Find S(0) and limq→∞S(q) and in each case explain what your findings mean in medical
3. terms.
4. (c) What is the optimal daily drug quantity to administer (in terms of k1 and k2) so as to maximize the fraction of patients surviving?
5. (d) Suppose Health Canada has only approved the use of the drug of up to 45 mg/day and sup- pose k1 = 25 mg/day is the same for all patients but k2 varies from patient to patient. To determine a personalized treatment strategy it would be useful for physicians to have a plot of the optimal daily drug quantity as a function of k2, call it q∗(k2). Sketch a plot of q∗(k2) and explain why you drew it that way. Hint: don’t forget that k1 < k2!
6. (e) Measuring k2 for a patient is imperfect. Which patients are more at risk for a significant dosing error if their measured k2 is a little off: patients with high k2 or low k2? Use your graph to explain the reasoning.

Answer 1

Hi Kyla S.,

I'll give you a few pointers and let you work from there.

First, look at the equation you supplied: for q=0 survival is k1 (spontaneous remission); as q->infinity survival-> infinity (the problem did state that k1, k2 are both positive, and q is a dosage, so positive).

That contradicts what the problem also states, that the drug is lethal at high dosages (as all drugs are!). So, the equation supplied (or some of the conditions) are incompatible, and the problem is either non-sense or incorrectly transcribed from the original source. I don't see an obvious fix (any single operator missing, for example).

That being the case, differentiating S by q, which would normally give you the answer for (4) as a "zero", wouldn't be of much use.

Suggest you check the web for a better survival equation and use it instead, stating why you did that.

What a shame, the answers sought are "right on".

-- Cheers, -- Mr. d.