Question

Suppose the dosage of a chemotherapeutic drug is proportional to tumour surface area, where the tumour is assumed to be spherical in shape. Initially, a dose of 100 IU (International Units) is administered to a patient to treat a tumour. After two weeks, the tumour diameter is observed to shrink by 30%. Compute the correct dose for a second round of chemotherapy with the same drug, to be administered two weeks after the initial dose.

Answer 1

So surface area is equal to 4(pi)r^2. To make the problem simple let's assume the radius is equal to 1. The given information shows that a treatment of 100 IU eliminates 30% of the diameter which is also 30% of the radius. This is not however 30% of the surface area. The surface area, assuming a initial radius of 1 unit, went from 4 surface units to 1.96 surface units.

S=4(pi)(1)^2=4(pi)
S'=4(pi)(.7)^2=1.96(pi)

So a treatment of 100 IU took out 2.04 surface units. If that took 100 IU then to find the dosage required for the rest a simple ratio equation could solve it.

2.04/1.96=100/x
x=96.0784313725 IU

This answer seems logical because you have almost the same amount of surface area so the dosage is close but the surface area left is slightly less than what was shrunk (1.96 is slight less than 2.04) so the dosage would be slightly less than 100 IU