Problem 57 - Calculus for Biology and Medicine

Question

You do not know whether a drug has zeroth order or first order elimination kinetics. You will use data to determine which type of kinetics it has. You measure the concentration of the drug (in \(\mathrm{mg} / \mathrm{ml}\) ) at time \(t=0\) and at time \(t=1\). No drug is added to the blood in this interval. You measure the following data: \begin{tabular}{ll} \hline \(\boldsymbol{t}\) & \(\boldsymbol{c}_{t}\) \\ \hline 0 & 50 \\ 1 & 35 \\ \hline \end{tabular} (a) Assume that the drug has zeroth order kinetics. What amount is eliminated from the blood each hour? (b) Assume that the drug has zeroth order kinetics and no more drug is added. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (c) Now assume the drug has first order elimination kinetics. What percentage of drug is eliminated from the blood each hour? (d) Assume that the drug has first order kinetics and no more drug is added to the blood. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\) (e) You measure the concentration at time \(t=2\) and find \(c_{2}=\) 20\. By comparing with your predictions from (b) and (d), decide: Does the drug have zeroth or first order kinetics?

Step-by-Step Solution

Answer

The drug exhibits zeroth order kinetics, as confirmed by the matching prediction for \(c_2\).

Step 1: Zeroth Order Kinetics Elimination Rate

In zeroth order kinetics, the elimination of the drug occurs at a constant rate. Given the data for time \( t=0 \) and \( t=1 \), you can find the rate of elimination by calculating the difference between the concentrations \[Rate = c_{0} - c_{1} = 50 - 35 = 15 \text{ mg/ml per hour}\]This means 15 mg/ml of the drug is eliminated from the blood each hour.

Step 2: Zeroth Order Kinetics Recursion Relation

For zeroth order kinetics, the concentration at time \( t \) is given by the formula:\[c_{t} = c_{t-1} - ext{Rate}\]Substituting the rate found in Step 1, the recursion relation is:\[c_{t} = c_{t-1} - 15\]Predicting \(c_{2}\) (concentration at \(t=2\)):\[c_{2} = c_{1} - 15 = 35 - 15 = 20 \]

Step 3: First Order Kinetics Elimination Percentage

In first order kinetics, the drug is eliminated at a rate proportional to its concentration. The percentage of drug eliminated per hour is found by:\[\text{Percentage} = \frac{c_{0} - c_{1}}{c_{0}} \times 100\%\]Substituting the given values:\[\text{Percentage} = \frac{50 - 35}{50} \times 100 = 30\%\]This means 30% of the drug is eliminated from the blood each hour.

Step 4: First Order Kinetics Recursion Relation

The relation for first order kinetics can be described using the formula:\[c_{t} = c_{t-1} \times (1 - k)\]where \(k\) is the proportion of drug eliminated each hour. Given \(k=0.3\), the recursion relation becomes:\[c_{t} = c_{t-1} \times 0.7\]Predicting \(c_{2}\) (concentration at \(t=2\)):\[c_{2} = c_{1} \times 0.7 = 35 \times 0.7 = 24.5\]

Step 5: Compare Predictions to Actual Measurement

From predictions, when comparing the predicted \(c_{2}\) values, the zeroth order prediction \( (c_{2} = 20) \) matches the measured value, whereas the first order prediction didn't. Therefore, the drug follows zeroth order kinetics.

Key Concepts

Zeroth Order Kinetics

In zeroth order kinetics, the drug is eliminated from the bloodstream at a constant rate. Unlike other kinetics models, the rate of elimination does not depend on the concentration of the drug. This means that every hour, the same amount of drug is cleared from the blood, regardless of how much drug is present initially. For example, if 15 mg/ml is the rate of elimination, that exact amount will be removed from the system each hour.

• The rate of elimination is constant.
• Independent of drug concentration.
• Amount of drug removed is the same every hour.

Understanding this model is crucial for specific pharmaceutical applications where a steady elimination is expected or required.

First Order Kinetics

First order kinetics describe a situation where the rate of drug elimination is directly proportional to the concentration of the drug in the blood. As such, a higher concentration results in a faster rate of elimination. This means that the percentage of drug eliminated remains constant over time. Hence, if 30% of the drug is removed each hour, that percentage applies regardless of initial concentration levels.

• Elimination rate is proportional to concentration.
• A constant percentage of the drug is cleared per time unit.
• Results in a decrease that becomes slower over time as concentration decreases.

Such models are common in many drug metabolism scenarios where initial drug concentration significantly influences clearance.

Recursion Relations

Recursion relations help predict future concentrations based on past data. In the context of drug kinetics, they provide a simple iterative method to understand how drug concentration changes over time.

Zeroth Order Recursion

For zeroth order kinetics:

• The formula is: \(c_{t} = c_{t-1} - ext{Rate}\).
• Every subsequent concentration is calculated by subtracting the constant rate.
• This explicit relation makes it easy to determine future concentrations quickly.

First Order Recursion

For first order kinetics:

• The formula is: \(c_{t} = c_{t-1} \times (1-k)\).
• Each new concentration is the previous concentration reduced by a consistent percentage.
• It accounts for diminishing returns in elimination as concentrations drop over time.

Elimination Rate

Elimination rate refers to how quickly a drug is removed from the bloodstream. It plays a critical role in determining appropriate dosing and understanding drug behavior in the body.

Zeroth Order Elimination Rate

• Is constant, meaning a set amount of drug is removed per unit time.
• Calculated as the difference between successive concentrations over time intervals.
• Can be seen in drugs that are eliminated at a fixed pace, often due to saturation of metabolic pathways.

First Order Elimination Rate

• Varies as it is a percentage of the current concentration.
• Reflects the exponential nature of decay, applicable to most pharmaceuticals under normal conditions.
• Is calculated as the percentage of the concentration change relative to the starting concentration.

Understanding these rates helps in crafting effective therapeutic regimens and in making informed medical and pharmacological decisions.