Question

A fixed dose of a given drug increases the concentration of that drug above normal levels in the bloodstream by an amount \(C_{0}\) (measured in percent). The effect of the drug wears off over time such that the concentration at some time \(t\) is \(C_{0} e^{-k t}\) where \(k\) is the known rate at which the concentration of the drug in the bloodstream declines. (a) Find the residual concentration \(R\), the accumulated amount of the drug above normal levels in the bloodstream, at time \(t\) after \(n\) doses given at intervals of \(t_{0}\) hours starting with the first dose at \(t=0\). (b) If the drug is alcohol and 1 oz. of alcohol has \(C_{0}=0.05 \%\), how often can a "dose" be taken so that the residual concentration is never more than \(0.15 \%\) ? Assume \(k=(1 / 3) \ln (2)\)

Short answer

The interval between doses should be at least 1.71 hours to keep the residual concentration under 0.15%.

Step-by-step solution

Understanding the Residual Concentration Formula

The residual concentration \(R\) after \(n\) doses given at intervals of \(t_{0}\) hours is the sum of the effects of each dose. The concentration effect of each dose \(i\) after time \(t = n t_0\) is \(C_{0} e^{-k (n-i)t_{0}}\). Thus, the total residual concentration is the sum of all these effects: \[R = \sum_{i=0}^{n-1} C_{0} e^{-k (n-i)t_{0}} = C_{0} \sum_{i=0}^{n-1} e^{-k i t_{0}}\] This is a geometric series with ratio \(e^{-k t_{0}}\).

Calculate the Sum of the Geometric Series

To find the sum of the series, use the formula for the sum of a geometric series: \[\text{Sum} = a \frac{1-r^{n}}{1-r}\] where \(a = C_{0}\) and \(r = e^{-k t_{0}}\). Therefore, \[R = C_{0} \frac{1-e^{-k n t_{0}}}{1-e^{-k t_{0}}}\] Since doses can be taken indefinitely, we consider the limit as \(n \rightarrow \infty\), which becomes: \[R = \frac{C_{0}}{1-e^{-k t_{0}}}\] This formula gives the accumulated residual concentration after many doses.

Set up the Residual Concentration Condition for Alcohol

Given that each dose of alcohol has \(C_{0}=0.05\%\), we want \(R \leq 0.15\%\). So, substituting in the concentration values, we get: \[\frac{0.05}{1-e^{-(1/3) \ln(2) t_{0}}} \leq 0.15\] This inequality constrains the time interval \(t_{0}\) between doses.

Solve the Inequality for \(t_{0}\)

Rearranging and solving the inequality: \[0.05 \leq 0.15 (1-e^{-(1/3) \ln(2) t_{0}})\] Simplify to get: \[\frac{1}{3} \leq 1-e^{-(1/3) \ln(2) t_{0}}\] \[e^{-(1/3) \ln(2) t_{0}} \leq \frac{2}{3}\] Take the natural logarithm: \[-(1/3) \ln(2) t_{0} \leq \ln\left(\frac{2}{3}\right)\] \[t_{0} \geq \frac{3 \ln \left(\frac{3}{2}\right)}{\ln(2)}\]

Calculate the Minimum Interval \(t_{0}\)

Computing the numerical value: \[t_{0} \geq \frac{3 \ln(1.5)}{\ln(2)} \approx 1.71 \text{ hours}\] Therefore, the interval between doses should be at least approximately 1.71 hours.

Key concepts

Geometric Series

When you have a repeating process over time, like taking a drug at specific intervals, the series of outcomes can form a "geometric series." A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number, called the "common ratio."

In the context of drug concentration, after each dose, the concentration decreases exponentially. When calculating the total concentration after multiple doses, you sum up these exponentially decreasing effects, forming a geometric series.

• The first term ( \( a \) ) is the initial effect from the first dose, given by the formula \( C_0 \).
• The common ratio ( \( r \) ) is the decay factor given by \( e^{-k t_0} \).

By using the formula for summing a geometric series, \( \frac{a (1-r^n)}{1-r} \), you can find the residual concentration, \( R \), after many doses. If doses are taken indefinitely, \( n \rightarrow \infty \), the series converges, and the sum becomes simpler: \( R = \frac{C_0}{1 - r} \). This demonstrates how powerful geometric series are in modeling cumulative effects over time.

Drug Concentration

With drugs, the concentration in the bloodstream is crucial for effectiveness and safety. Drugs generally get absorbed into the bloodstream and then gradually reduce in concentration over time through metabolic processes. This drop in concentration follows an exponential pattern, which means it decreases rapidly at first and then slows down.

The formula for the drug concentration at any time \( t \) is \( C_0 e^{-kt} \). Here:

• \( C_0 \) represents the initial increase in concentration due to the drug dose.
• \( k \) is the rate constant that depicts how fast the drug concentration decreases.
• \( t \) is the time elapsed since the dose.

Understanding how drug concentration behaves helps in determining the timing and dosage of administration. It's important for ensuring therapeutic levels are maintained without reaching toxic levels, particularly in repetitive dosing like in the case of alcohol or other substances.

Mathematical Modeling

Mathematical modeling is a way to describe real-world situations using mathematical expressions and equations. It simplifies complex phenomena and helps predict future outcomes. In this exercise, mathematical modeling of drug concentration helps determine a safe dosing schedule.

The process involves:

• Identifying key variables, like drug concentration \( C_0 \), decay rate \( k \), and time intervals \( t_0 \).
• Using known formulas, such as the exponential decay and geometric series, to model how drug residuals accumulate with repeated doses.
• Setting conditions or constraints, like limiting the residual concentration to avoid adverse effects.

For instance, by modeling alcohol's effect and setting an upper limit for safe concentration, one can solve for the minimum time interval between doses to maintain safety. Mathematical modeling thus becomes an essential tool in optimizing drug administration schedules and ensuring patient safety.