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. Give all answers below to at least 3 significant figures, including the times of effectiveness, time of maximum dose, and value of maximum dose.
a. The simplest situation calls for an injection of the drug into the body. In this case, a fixed amount of the drug enters the body, then its quantity decreases exponentially as the drug is metabolized and excreted from the body. Suppose that for a certain patient, it is found that the amount of a certain drug in his body satisfies the equation
where your patient has k = 0.03 (day-1). Determine how long the drug is effective if it has been determined that the patient must have at least 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. (One example of this can be found at the webpage for SkyePharma for their drug delivery system depofoamTM http://www.skyepharma.com/injectables_depofoam.html.) This delivery system can prevent large toxic doses in the body and maintain the drug level for longer at theurapeutic doses. Suppose that the amount of drug delivered by this new type of drug delivery system satisfies the model
where B0 = 14.3 (mg), q = 0.1 (day-1), and p = 0.03 (day-1). The first decaying exponential is from the body metabolism, while the second one is from the polymer degradation. (It can be shown that this is the same amount of drug as delivered in Part a.) Once again assume that the patient must have 3 mg in his body to be effective. 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, A(t) and B(t), 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?