Problem 56
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 between \(t=0\) and \(t=1\). You measure the following data: \begin{tabular}{ll} \hline \(\boldsymbol{t}\) & \(\boldsymbol{c}_{t}\) \\ \hline 0 & 40 \\ 1 & 32 \\ \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 to the blood. 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. Write a recursion relation for \(c_{t}\) and predict \(c_{2}\). (e) You measure the concentration at time \(t=2\) and find \(c_{2}=\) 25.6. By comparing with your predictions from (b) and (d), decide: Does the drug have zeroth or first order kinetics?
Step-by-Step Solution
Verified Answer
The drug exhibits first order kinetics, as the predictions match the measured data.
1Step 1: Zeroth Order Kinetics Elimination Rate
For zeroth order kinetics, the rate of elimination is constant over time. Given the concentrations at \(t=0\) and \(t=1\), we calculate this constant rate.The concentration decreases from 40 to 32 \(\text{mg/ml}\) in one hour. Thus, the rate of elimination is \(40 - 32 = 8 \text{ mg/ml/hr}\).
2Step 2: Zeroth Order Recursion Relation and Prediction
For zeroth order kinetics, the concentration at time \(t+1\) is expressed as \(c_{t+1} = c_t - k\), where \(k\) is the rate of elimination.Given \(c_t = 32\) and \(k = 8\), for \(t=2\) we predict: \[ c_2 = c_1 - 8 = 32 - 8 = 24 \text{ mg/ml} \].
3Step 3: First Order Kinetics Elimination Percentage
First order kinetics involve a constant proportion of drug elimination per hour.The fractional change in concentration from \(t=0\) to \(t=1\) is:\[ \frac{c_1}{c_0} = \frac{32}{40} = 0.8 \].This implies \(20\%\) of the drug is eliminated per hour, as \(1 - 0.8 = 0.2\).
4Step 4: First Order Recursion Relation and Prediction
For first order kinetics, the recursion relation is \(c_{t+1} = c_t \times (1-r)\), where \(r\) is the percentage eliminated per hour (in decimal).With \(r = 0.2\), for \(t=2\) we have:\[ c_2 = c_1 \times 0.8 = 32 \times 0.8 = 25.6 \text{ mg/ml} \].
5Step 5: Comparing Predictions with Measured Data
Measured concentration at \(t=2\) is \( 25.6 \text{ mg/ml} \).- Zeroth order prediction was \(24 \text{ mg/ml}\).- First order prediction was \(25.6 \text{ mg/ml}\).Since the measured value of \(c_2\) matches exactly with the first order prediction, the drug follows first order kinetics.
Key Concepts
Zeroth order kineticsFirst order kineticsDrug concentration measurement
Zeroth order kinetics
Zeroth order kinetics in pharmacology implies that a drug is eliminated from the bloodstream at a consistent and constant rate, irrespective of its concentration. This means the amount of drug removed per unit time remains unchanged, and does not depend on how much drug is present at the time of measurement.
In practical terms, if you start with a drug concentration of 40 mg/ml and one hour later it reduces to 32 mg/ml, it indicates 8 mg/ml has been consistently eliminated in that timeframe. This rate of 8 mg/ml per hour will continue as long as the drug elimination follows zeroth order kinetics.
By recognizing zeroth order kinetics, we can predict future concentrations. For instance, given an elimination rate of 8 mg/ml, if the concentration at one hour is 32 mg/ml, the concentration at two hours will expectedly be 24 mg/ml. The equation to model this is straightforward: \( c_{t+1} = c_t - k \), where \( k \) is the rate of elimination, being 8 mg/ml/hr in this example.
In practical terms, if you start with a drug concentration of 40 mg/ml and one hour later it reduces to 32 mg/ml, it indicates 8 mg/ml has been consistently eliminated in that timeframe. This rate of 8 mg/ml per hour will continue as long as the drug elimination follows zeroth order kinetics.
By recognizing zeroth order kinetics, we can predict future concentrations. For instance, given an elimination rate of 8 mg/ml, if the concentration at one hour is 32 mg/ml, the concentration at two hours will expectedly be 24 mg/ml. The equation to model this is straightforward: \( c_{t+1} = c_t - k \), where \( k \) is the rate of elimination, being 8 mg/ml/hr in this example.
First order kinetics
First order kinetics is a different ball game compared to zeroth order kinetics. Here, the drug is eliminated at a rate proportional to its current concentration. Instead of a constant amount, a constant *percentage* of the remaining drug is cleared per hour.
To illustrate, if your initial drug concentration is 40 mg/ml, and it drops to 32 mg/ml after one hour, this removal represents a fractional reduction. In percentage terms, this change is calculated as: \( \frac{32}{40} = 0.8 \) or 80% of the drug remains, thus indicating a 20% removal per hour.
When predicted using first order kinetics, if 20% of the drug is eliminated each hour, the recursion model will be \( c_{t+1} = c_t \times (1 - r) \), where \( r \) is the fractional removal rate (0.2 in this case). Applying this model at a concentration of 32 mg/ml gives: \( 32 \times 0.8 = 25.6 \) mg/ml. This matches measured data, confirming first order kinetics.
To illustrate, if your initial drug concentration is 40 mg/ml, and it drops to 32 mg/ml after one hour, this removal represents a fractional reduction. In percentage terms, this change is calculated as: \( \frac{32}{40} = 0.8 \) or 80% of the drug remains, thus indicating a 20% removal per hour.
When predicted using first order kinetics, if 20% of the drug is eliminated each hour, the recursion model will be \( c_{t+1} = c_t \times (1 - r) \), where \( r \) is the fractional removal rate (0.2 in this case). Applying this model at a concentration of 32 mg/ml gives: \( 32 \times 0.8 = 25.6 \) mg/ml. This matches measured data, confirming first order kinetics.
Drug concentration measurement
Understanding and accurately measuring drug concentration is crucial for determining the kinetic order of drug elimination. The concentration at different times allows us to discern the pattern of drug clearance from the body.
In the given study, concentrations at \( t=0 \) and \( t=1 \) were first recorded to observe changes in drug levels: going from 40 mg/ml to 32 mg/ml. This measurement is pivotal, offering a basis to assess if consistent amounts or proportions are being eliminated.
Moreover, further measurement, such as at \( t=2 \), aids in validating hypotheses about first or zeroth order kinetics. Like seeing if the result aligns with a predicted model whether it’s \( c_2 = 24 \) mg/ml for zeroth order or \( c_2 = 25.6 \) mg/ml for first order. When lab values coincide with a particular model, it asserts that model's applicability for the specific drug under study.
In the given study, concentrations at \( t=0 \) and \( t=1 \) were first recorded to observe changes in drug levels: going from 40 mg/ml to 32 mg/ml. This measurement is pivotal, offering a basis to assess if consistent amounts or proportions are being eliminated.
Moreover, further measurement, such as at \( t=2 \), aids in validating hypotheses about first or zeroth order kinetics. Like seeing if the result aligns with a predicted model whether it’s \( c_2 = 24 \) mg/ml for zeroth order or \( c_2 = 25.6 \) mg/ml for first order. When lab values coincide with a particular model, it asserts that model's applicability for the specific drug under study.
Other exercises in this chapter
Problem 55
A drug has first order elimination kinetics. At time \(t=0\) an amount \(a_{0}=20 \mathrm{mg}\) is present in the blood. One hour later, at \(t=1\), an amount \
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Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists
View solution Problem 56
Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists
View solution Problem 57
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 t
View solution