Problem 51 - Calculus for Biology and Medicine

Question

Adderall is a proprietary combination of amphetamine salts that is used to treat ADHD (AttentionDeficit/Hyperactivity Disorder). The patient takes one pill before \(8 \mathrm{am}\), and the drug has completely entered the blood by \(8 \mathrm{am} .\) At 8 am the blood concentration of the drug is \(33.8 \mathrm{ng} / \mathrm{ml}\). Adderall has first order elimination kinetics with \(7.7 \%\) of the drug being removed from the blood in each hour. The data in this equation are taken from Greenhill et al. (2003). (a) The concentration of drug in the patient's blood \(t\) hours after 8 am is measured to be \(C_{t}\). Write a recursive relation for \(C_{t+1}\) in terms of \(C_{t}\). Assume that the patient takes no other Adderall pills. (b) Solve your recurrence equation to derive an explicit formula for \(C_{t}\) as a function of \(t\). (c) When does the concentration of drug first drop below \(0.1 \mathrm{ng} / \mathrm{ml} ?\)

Step-by-Step Solution

Answer

The drug concentration first drops below 0.1 ng/ml at 58 hours after 8 am.

Step 1: Understanding First Order Elimination

First-order elimination kinetics implies that a constant percentage of the drug is eliminated per unit time. Here, it is given that 7.7% of the drug is removed each hour. Thus, the remaining concentration after each hour is 92.3% of the concentration of the previous hour.

Step 2: Defining the Recursive Relation

Given the concentration of drug at time \(t\), denoted as \(C_t\), the concentration at time \(t+1\), denoted as \(C_{t+1}\), can be defined using the remaining percentage: \[C_{t+1} = C_t \times 0.923\] This recursive relation reflects the 7.7% reduction in the drug's concentration per hour.

Step 3: Solving the Recurrence Relation

To find an explicit formula for \(C_t\), substitute the recursive formula repeatedly:\[\\begin{align*}C_1 &= C_0 \times 0.923, \C_2 &= C_1 \times 0.923 = C_0 \times 0.923^2, \&\vdots \C_t &= C_0 \times 0.923^t\end{align*}\]Thus, the explicit formula is: \[C_t = 33.8 \times 0.923^t\] where \(C_0 = 33.8 \text{ ng/ml}\) at \(t=0\).

Step 4: Finding the Time when Concentration Drops Below 0.1 ng/ml

Set \(C_t < 0.1\) and substitute the explicit formula:\[33.8 \times 0.923^t < 0.1\]Solving for \(t\), divide both sides by 33.8:\[0.923^t < \frac{0.1}{33.8}\]Taking the logarithm of both sides:\[t \ln(0.923) < \ln\left(\frac{0.1}{33.8}\right)\]Solve for \(t\):\[t > \frac{\ln\left(\frac{0.1}{33.8}\right)}{\ln(0.923)} \approx 57.3\] Rounded up, since \(t\) must be an integer, the concentration first drops below 0.1 ng/ml at \(t = 58\) hours.

Key Concepts

Recursive Relation in Drug Concentration

When analyzing the concentration of a drug like Adderall in the bloodstream, a key concept is the recursive relation, which helps predict the future concentration based on its current state. Think of a recursive relation as a mathematical way of expressing, "If I know the concentration now, how can I predict the concentration in one hour?"
A recursive relation establishes a link between successive values. Here, it is represented by the equation:

• \[C_{t+1} = C_t \times 0.923\]

This equation tells us that the concentration at time \(t+1\) is equal to the concentration at time \(t\) multiplied by 0.923. The factor 0.923 comes from the fact that 7.7% of the drug is eliminated each hour, leaving behind 92.3%. By understanding and utilizing this relation, you can track how the concentration decreases over time as the body processes the drug.
Recursive relations are powerful because they break down complex processes into simple, predictable steps. Each step only requires knowledge of the step before it, creating a straightforward way to model change over time. This foundation is essential as it builds up to more comprehensive formulas.

The Explicit Formula for Determining Drug Levels

While recursive relations provide a step-by-step approach, sometimes it's helpful to jump straight to the answer for any given hour. That's where the explicit formula comes into play. An explicit formula allows you to directly compute the concentration at any time \(t\), without having to calculate all preceding concentrations.
The explicit formula derived from this scenario is:

• \[C_t = 33.8 \times 0.923^t\]

In this formula, \(33.8\) represents the initial concentration of the drug at \(t=0\). The term \(0.923^t\) accounts for the exponential decrease in concentration due to the continuous 7.7% elimination rate. As \(t\) increases, \(0.923^t\) grows smaller, signifying a further decline in the drug concentration.
Using this direct approach, you can plug in any value of \(t\) to find the concentration without needing to manually compute each prior hour's concentration. This feature makes the explicit formula incredibly efficient and useful for scenarios that span extended periods.

Understanding Drug Concentration Dynamics

Drug concentration is a critical factor in understanding the effectiveness and safety of medications. Here, we're focusing on the concentration of Adderall, measured in nanograms per milliliter (ng/ml), as it diminishes over time due to the body's elimination processes.
The initial concentration starts at 33.8 ng/ml. Over time, due to first-order elimination kinetics, the concentration decreases exponentially. First-order kinetics means that the rate of drug elimination is proportional to its current concentration. So, as the drug level decreases, the amount eliminated per unit time reduces, giving an exponential decay in concentration.

• Each hour, 7.7% of the drug is removed, which can also be expressed as 92.3% of the previous amount remaining.
• The concentration after one hour is calculated by:\(C = C_0 \times 0.923\), where \(C_0\) is the initial concentration.

Understanding these properties helps predict how long it will take for the drug to fall below a certain level, like 0.1 ng/ml in this scenario. This knowledge is vital for determining dosing schedules and ensuring therapeutic—but safe—drug levels are maintained in patients.