Problem 9 - Calculus for Biology and Medicine

Question

Concentration of Adderall in Blood The drug Adderall (a proprietary combination of amphetamine salts) is used to treat ADHD. Adderall has first order kinetics for elimination, with elimination rate constant \(k_{1}=0.08 \mathrm{hr}\). We will assume that pills are quickly absorbed into the blood; that is, when the patient takes a pill their blood concentration of the drug immediately jumps. (a) Assuming that the patient takes one pill at \(8 \mathrm{am}\), the concentration in their blood after taking the pill is \(33.8 \mathrm{ng} / \mathrm{ml}\). Assuming that they take no other pills during the day, write down and then solve the differential equation that gives the concentration of drug in the blood, \(M(t)\), over the course of a day. (Hint: You may find it helpful to define time \(t\) by the number of hours elapsed since \(8 \mathrm{am} .)\) (b) What is the blood concentration just before the patient takes their next dose of the drug, at \(8 \mathrm{am}\) the next day? (c) At what time during the day does the blood concentration fall to half of its initial value? (d) In an alternative treatment regimen, the patient takes two half pills, one every 12 hours. They take the first pill at \(8 \mathrm{am}\), with no drug in their system. Why would we expect their drug concentration to be \(16.9 \mathrm{ng} / \mathrm{ml}\) immediately after taking the pill? (e) What is the concentration in their blood at \(8 \mathrm{pm} ?\) (f) At \(8 \mathrm{pm}\), the patient takes the other half pill. This increases the concentration of Adderall in their blood by \(16.9 \mathrm{ng} / \mathrm{ml}\) (i.e.. the concentration increases at \(8 \mathrm{pm}\) by \(16.9 \mathrm{ng} / \mathrm{ml}\) ). Derive a formula for the concentration in their blood as a function of time elapsed since \(8 \mathrm{am}\). (You will need different expressions for the concentration for \(0

Step-by-Step Solution

Answer

The solutions are: (b) ~5.02 ng/ml, (c) ~4:40 pm, (h) ~4.93 ng/ml.

Step 1: Define the Differential Equation

The elimination of Adderall follows first-order kinetics, modeled by the differential equation \( \frac{dM}{dt} = -k_1 M(t) \), where \( M(t) \) is the concentration of the drug at time \( t \), and \( k_1 = 0.08 \, \text{hr}^{-1} \) is the elimination rate constant.

Step 2: Solve the Differential Equation

The general solution to \( \frac{dM}{dt} = -0.08 M \) is \( M(t) = M(0) e^{-0.08t} \). Given \( M(0) = 33.8 \, \text{ng/ml} \) at \( t = 0 \) (which is 8 am), the specific solution is \( M(t) = 33.8 e^{-0.08t} \).

Step 3: Concentration at Next Dose Time

Evaluate \( M(t) \) at \( t = 24 \) hours: \( M(24) = 33.8 e^{-0.08 \times 24} \). Calculating this gives \( M(24) \approx 5.02 \, \text{ng/ml} \).

Step 4: Half-life of the Drug

To find when the concentration falls to half its initial value, solve \( M(t) = \frac{33.8}{2} = 33.8 e^{-0.08t} \). Simplify and solve \( t = \frac{\ln(0.5)}{-0.08} \approx 8.66 \) hours, which is approximately 4:40 pm.

Step 5: Immediate Concentration from Half Pills

Since half pills are given, and each adds \( 16.9 \, \text{ng/ml} \), taking one half pill at 8 am with no drug initially leads to \( 16.9 \, \text{ng/ml} \) immediately after.

Step 6: Concentration at 8 pm Considering First Half Pill

Using \( M(t) = 16.9 e^{-0.08t} \), find \( M(12) = 16.9 e^{-0.08 \times 12} \approx 4.93 \, \text{ng/ml} \).

Step 7: Formula for Concentration Post 8 pm Dose

After 8 pm, the concentration is the sum of remaining concentration from the first half pill and the immediate increase from the second half pill. For \(12 < t < 24 \), \( M(t) = 16.9 e^{-0.08(t-12)} + 16.9 \).

Step 8: Minimum Blood Concentration Over 24 Hours

The lowest concentration occurs just before the second dose at 8 pm, given by \( M(12) \approx 4.93 \, \text{ng/ml} \), when using the regimen with half pills.

Key Concepts

First-order kinetics

First-order kinetics describes a situation where the rate of a reaction or process is directly proportional to the concentration of a single reactant or component. This means that as the concentration decreases, the rate slows down proportionally. In our case, the elimination of Adderall follows this principle. The differential equation given by \( \frac{dM}{dt} = -k_1 M(t) \) models this relationship, where \( M(t) \) is the concentration of the drug at time \( t \) and \( k_1 \) is the elimination rate constant.

This equation indicates that the rate of change in drug concentration decreases exponentially over time. When solving this differential equation, one can see the drug concentration declining exponentially. This is typical for substances following first-order kinetics. Understanding this principle is essential for calculating how long drugs remain effective in the system.

Elimination rate constant

The elimination rate constant, denoted as \( k_1 \), is a crucial parameter in pharmacokinetics. It quantifies the rate at which a drug is removed from the body, typically having units of inverse time, such as \( \, \text{hr}^{-1} \).

For Adderall, \( k_1 \) is given as 0.08 \( \, \text{hr}^{-1} \). This constant helps determine how quickly the concentration of the drug decreases over time, which is critical for dosing schedules and predicting plasma concentration at different times. An important application of the elimination rate constant is in the solution of the first-order kinetic equation.

With this rate constant, we can calculate the drug's concentration at any time using the formula \( M(t) = M(0) e^{-k_1 t} \). This expression shows that the concentration declines exponentially, providing insights into how the drug behaves in the body over time.

Half-life calculation

The half-life of a substance is the time it takes for its concentration to decrease to half of its initial value. It is a significant concept in understanding drug dosing and how long a drug remains in the body.

For a first-order kinetics process, the half-life, \( t_{1/2} \), is calculated using the formula:
\[ t_{1/2} = \frac{\ln(2)}{k_1} \]
where \( \ln(2) \approx 0.693 \) is the natural logarithm of 2.

Using the given elimination rate constant for Adderall \( k_1 = 0.08 \, \text{hr}^{-1} \), the half-life can be calculated as:
\[ t_{1/2} = \frac{0.693}{0.08} \approx 8.66 \, \text{hours} \]
This calculation shows that it takes roughly 8 hours and 40 minutes for the concentration of Adderall to reduce by half. Knowing the half-life helps in planning proper dosing intervals to maintain the desired drug levels in the bloodstream.