3. Several of you are considering careers in medicine and biotechnology. Drug therapy and dose response is very important in the treatment of many diseases, particularly cancer. Since cancer cells are very similar to your normal body cells, their destruction relies on very toxic drugs. There are some very fine lines in certain cancer treatments between an ineffective dose, one that destroys the cancer, and one that is toxic to all cells in the body. At the base of many of the calculations for these treatments are simple mathematical models for drug uptake and elimination.
a. The simplest situation calls for an injection of the drug into the body. In this case, the differential equation describing the amount of drug in the body is given by:
where A0 is the amount of drug injected and k depends on how the drug is metabolized and excreted from the body. Let A(0) = 10 mg and assume that it is determined that your patient has k = 0.03 (day -1). Solve this differential equation and determine how long the drug is effective if it has been determined that the patient must have 3 mg in his body.
b. With new materials being developed, the drug can be inserted into polymers that slowly decay and release the drug into the body. This delivery system can prevent large toxic doses in the body and maintain the drug level for longer at theurapeutic doses. A differential equation that describes type of drug delivery system is given by
where r = 1.0 (mg/day) and q = 0.1 (day -1). (It can be shown that if r/q = A0, then this is the same amount of drug as delivered in Part a.) Solve this differential equation. Over what time period (if any) is this therapy effective. Is this time period longer or shorter than your answer from Part a.?
c. On a single graph show both solutions for 60 days. Find what the maximum dose is in the body from the second treatment given in Part b. and when this occurs. Which treatment do you consider to be superior and why?
d. For this part of the problem, we want to find the numerical solution of the differential equation in Part b, using the Improved Euler's method of the previous lab. Take a stepsize of h = 0.5 on the differential equation describing the drug delivery system with polymers and use the Improved Euler's method to simulate the differential equation for 0 < t < 60. Graph the numerical solution using the Improved Euler's method with the actual solution found in Part b, using Maple. Create a table with the values of the actual solution, the solution from Improved Euler's method, and the percent error between these solutions at times t = 10, 20, 30, 40, 50, 60. Does this numerical solution adequately represent the actual solution of the differential equation?