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# Derivatives and Rate of Change : Drug Elimination Rate

PROBLEM STATEMENT: The concentration in the blood resulting from a single dose of a drug decreases with time as the drug is eliminated from the body. In order to determine the exact pattern that the decrease follows, experiments are performed in which drug concentrations in the blood are measured at various times after the drug is administered. The data are then checked against a hypothesized function relating drug concentration to time.

At this point we make a simplifying assumption: namely, that when the drug is administered, it is diffused so rapidly throughout the blood stream that, for all practical purposes it reaches its fullest concentration instantaneously. Thus the concentration jumps from a low or zero level to a higher level in zero time.

The simplest function to hypothesize as a model of drug concentration is a linear one: that is, we might start by assuming that the concentration of a drug in the blood is a linear function of the time since the dose was administered.

Suppose a single dose of a certain drug is administered to a patient at time t = 0, and that the blood concentration was measured immediately thereafter, and again after four hours. The results of such experiments are give in table 1.

Table 1

DATA Experiment 1 Experiment 2

Concentration at time t = 0 1.0 mg/ml 1.5 mg/ml
Concentration after 4 hours 0.15 mg/ml 0.75 mg/ml

A. For this part assume that the function describing concentration of time is linear. Each data set in table 1 represents a different drug and a different initial dose. For each data set:

1) Sketch a graph of the concentration function, that is, graph the level of
Concentration vs. time. Assume concentrations are measured in milligrams
Per milliliter and time is measured in hours.

2) Predict the time when the blood becomes free of the drug, assuming that no

3) Describe the rate at which the drug is eliminated. Does the rate of elimination
Seem to depend upon any other quantity (e,g, level of concentration)?

4) Predict what the graph of concentration level vs time would look like if further
Doses of the drug were administered every six hours for forty-eight hours.

5) Predict what would happen to the concentration level of the drug if it were

B. Now assume that the rate at which the concentration is decreasing at time t is
Proportional to the concentration level at time t. This idea can be modeled by the
differential equation, namely:

dy/dt = -ky

Where y is the concentration of the drug in the blood at time t, and k is a constant. Using
The same data sets as in part A, solve the differential equation and answer questions 1 to 5 above.

#### Solution Summary

A differential equation is solved.

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