Problem 62 - Calculus for Biology and Medicine

Question

Because of complex interactions with other drugs, some drugs have zeroth order elimination kinetics in some circumstances, and first order kinetics in other circumstances, depending on what other drugs are in the patient's system, as well as on age and preexisting medical conditions. Use the data on how concentration varies with time to determine whether the drug has zeroth or first order kinetics. Given the following sequence of measurements for drug concentration, determine whether the drug has zeroth or first order kinetics. $$ \begin{array}{lcccc} \hline \boldsymbol{t} \text { (Hours) } & 0 & 2 & 4 & 6 \\ \hline \boldsymbol{c}_{t}(\mathbf{m g} / \mathbf{m l}) & 40 & 36 & 32.4 & 29.16 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The drug follows first order kinetics.

Step 1: Understand Zeroth and First Order Kinetics

Zeroth order kinetics has a constant rate of elimination, leading to a linear decrease in concentration over time. In contrast, first order kinetics has a rate of elimination proportional to the concentration, resulting in an exponential decrease.

Step 2: Calculate Differences for Zeroth Order Kinetics

Check if the concentration decreases by a constant amount each time interval, indicating zeroth order kinetics. Calculate: - From 0 to 2 hours: 40 - 36 = 4 - From 2 to 4 hours: 36 - 32.4 = 3.6 - From 4 to 6 hours: 32.4 - 29.16 = 3.24 The differences are not constant, suggesting not zeroth order.

Step 3: Verify with First Order Kinetics Assumption

For first order kinetics, the ratio of concentrations over equal time intervals should remain constant. Calculate the ratios:- \(\frac{36}{40}\approx 0.9\)- \(\frac{32.4}{36}\approx 0.9\)- \(\frac{29.16}{32.4}\approx 0.9\)The ratios are approximately constant, supporting first order kinetics.

Step 4: Conclusion

Since the concentration decreases exponentially with time and the ratio of concentrations over time remains consistent, the drug follows first order kinetics. This conclusion is reached by refuting the hypothesis of zeroth order kinetics and verifying the characteristics of first order kinetics.

Key Concepts

Zeroth Order Kinetics

Zeroth order kinetics describes a situation where the rate of drug elimination from the body is constant, regardless of the drug concentration present. This means that no matter how high or low the concentration of the drug, it leaves the body at the same steady rate. This results in a linear decrease in drug concentration over time.
Imagine filling a bathtub with water. If you let the water out at a constant speed by opening the drain just a little, the water level will fall steadily at the same rate, which is akin to zeroth order kinetics:

• The concentration decreases by a fixed amount every hour.
• The graphical plot of concentration vs. time is a straight line.
• This scenario typically occurs when the drug elimination pathway is saturated.

In the exercise, the concentration did not decrease by a constant amount, indicating that zeroth order kinetics does not apply in this case.

First Order Kinetics

First order kinetics is much more commonly encountered in pharmacokinetics. Unlike zeroth order, in first order kinetics, the rate at which the drug is eliminated is proportional to the drug's current concentration. This means the higher the concentration, the faster the drug is removed. As a result, the concentration decreases exponentially rather than linearly.
Think about draining that same bathtub but with the stopper completely open this time. The initial water leaves quickly but slows as the level drops. This represents first order kinetics:

• The proportional decrease implies a constant fraction of drug is eliminated per unit time.
• The concentration-time curve is a downward sloping exponential curve.
• Most drugs follow this kinetic model under normal physiological conditions.

In the exercise, the fact that the ratio of concentrations between time intervals remained consistent confirms first order kinetics for the drug.

Drug Concentration Measurements

Measuring drug concentration over time is crucial in understanding how a drug is metabolized and eliminated. These measurements help determine whether a drug follows zeroth or first order kinetics, guiding appropriate dosing and treatment schedules.
In the exercise, measurements were given at several intervals:

• Initial concentration was highest at time zero.
• Each subsequent measurement displayed a lower concentration.
• Consistency in the calculation ratios pointed towards first order kinetics.

Drug concentration measurements often require precise laboratory techniques to ensure accuracy, particularly because small differences in concentration can significantly affect the kinetic analysis and subsequent medical decisions. This reliance on accurate data illustrates the importance of rigorous measurement practices in pharmacology.