Problem 52 - Calculus for Biology and Medicine

Question

Model painkillers that are absorbed into the blood from a slow release pill. Ourmathematical model for the amount, \(a_{t}\), of drug in the blood t hours after the pill is taken must include the amount absorbed from the pill each hour. Our model starts with the word equation. $$ \begin{array}{c} a_{t+1}=a_{t}+\begin{array}{l} \text { amount absorbed } \\ \text { from the pill } \end{array}-\begin{array}{l} \text { amount eliminated } \\ \text { from the blood } \end{array} \end{array} $$ Assume the amount absorbed from the pill between time \(t\) and time \(t+1\) is \(10 \cdot(0.4)^{t}\). (a) The drug has first order elimination kinetics. \(10 \%\) of the drug is eliminated from the blood each hour. Write down the recursion relation for \(a_{t+1}\) in terms of \(a_{t}\) (b) Assuming that \(a_{0}=0\), meaning that no drug is present in the blood initially, calculate the amount of drug present at times \(t=1,2, \ldots, 6\) (c) What is the maximum amount of drug present at any time in this interval? At what time is this maximum amount reached? (d) Use a spreadsheet to calculate the amount of drug present in hourly intervals from \(t=0\) up to \(t=24\). (e) Show when \(t\) is large, the amount of drug present in the blood decreases approximately exponentially with \(t .\) Hint: Plot the values that you computed for \(a_{t}\) against \(t\) on semilogarithmic axes.

Step-by-Step Solution

Answer

The recursion relation is \(a_{t+1} = a_t + 10 \cdot (0.4)^t - 0.1 \times a_t\). The maximum drug amount \(\approx 14.968\) is at \(t=6\). The drug decreases exponentially for large \(t\).

Step 1: Understand the Equation Components

To solve the given problem, first identify the components in the recursion relation for the drug amount in the blood: \(a_{t+1} = a_{t} + \text{amount absorbed} - \text{amount eliminated}\). This equation estimates the drug level each hour after considering absorption into the blood and elimination from the blood.

Step 2: Define Absorption and Elimination

The absorption from the pill is given as \(10 \cdot (0.4)^t\). The elimination is \(10\%\) of the drug in the blood after each hour. Thus, \(\text{amount eliminated} = 0.1 \times a_t\).

Step 3: Formulate the Recursion Relation

Substitute the absorption and elimination values into the recursion relation: \[a_{t+1} = a_t + 10 \cdot (0.4)^t - 0.1 \times a_t\]This equation will help calculate the drug amount in the blood at each time step.

Step 4: Calculate Drug Amount for (b)

Given that \(a_0 = 0\), calculate for the subsequent hours: - \(t=1\): \(a_1 = 0 + 10 \cdot (0.4)^0 - 0.1 \times 0 = 10\) - \(t=2\): \(a_2 = 10 + 10 \cdot (0.4)^1 - 0.1 \times 10 = 13\) - \(t=3\): \(a_3 = 13 + 10 \cdot (0.4)^2 - 0.1 \times 13 = 14.3\) - \(t=4\): \(a_4 = 14.3 + 10 \cdot (0.4)^3 - 0.1 \times 14.3 = 14.78\) - \(t=5\): \(a_5 = 14.78 + 10 \cdot (0.4)^4 - 0.1 \times 14.78 = 14.902\) - \(t=6\): \(a_6 = 14.902 + 10 \cdot (0.4)^5 - 0.1 \times 14.902 = 14.9678\)

Step 5: Find Maximum Amount and Time for (c)

Based on the calculations from Step 4, the maximum amount of drug occurs at about \(t=6\) with \(a_6 \approx 14.968\). This trend suggests the drug level reaches a plateau around this time.

Step 6: Use Spreadsheet for (d)

Using a spreadsheet, input the formula from Step 3 and initialize at \(a_0 = 0\). Calculate \(a_t\) from \(t=0\) to \(t=24\). Plot the values in your spreadsheet software to see how the drug amount changes over time.

Step 7: Examine Exponential Decrease for (e)

Plot \(a_t\) against \(t\) on semilogarithmic axes from your spreadsheet data. With \(t\) large, the plot should show an exponential decrease, confirming the approximation suggested.

Key Concepts

Recursion Relation

The term "recursion relation" might sound complex, but it's simply a mathematical way to predict future values based on current values. In our context, it's used to estimate the amount of drug in the bloodstream at the next time interval, depending on the current amount and some changes. The equation to represent this is:

• \[ a_{t+1} = a_{t} + \text{amount absorbed} - \text{amount eliminated}\]

Here, \(a_{t+1}\) represents the drug amount at the next hour, and \(a_{t}\) is the current amount. To model the absorption and elimination, we need additional information:

• The absorption from the pill for each hour is given by \(10 \cdot (0.4)^t\), which decreases with time as the pill's impact starts to diminish.
• We also have first-order kinetics for elimination where 10% of the drug, the current amount \(a_{t}\), is eliminated every hour. This is modeled as \(0.1 \times a_t\).

By plugging these into our core equation, we derive the specific recursion relation:

• \[ a_{t+1} = a_t + 10 \cdot (0.4)^t - 0.1 \times a_t \]

This relation helps us computationally track how the drug amount changes over time.

Exponential Decay

Exponential decay is a fascinating concept where quantities decrease proportionally over time. In the context of our drug model, it explains how the drug amount in the bloodstream changes as time progresses. The absorption decrease has an exponential character due to the factor \((0.4)^t\). This means that as time \(t\) increases, the absorbed amount diminishes rapidly in an exponential pattern. For elimination, since it's based on a fixed percentage per hour, it also displays exponential decay.

• This type of decay can be visualized clearly when we plot the logarithm of the drug amount against time.
• When you analyze such a plot (semilogarithmic), even visually, you can often detect an exponential curve if it appears as a straight line.

The result is that over a long period, the amount of drug doesn't just decrease slowly but at an accelerating exponential rate. This provides insights into how long-lasting a drug's effects can be and when re-administration might be necessary.

First-Order Kinetics

First-order kinetics is a common model in pharmacology for how drugs are processed by the body. It signifies that a constant proportion of the drug is eliminated over each time interval. In our exercise, we observe first-order kinetics through the fact that 10% of the drug present in the bloodstream is eliminated every hour. This is mathematically represented in our equation as \(0.1 \times a_t\).

• This model is useful because many biological processes, including drug metabolism, naturally tend to follow such a pattern.
• It reflects the reality that the more drug there is, the more gets eliminated, but as the drug amount decreases, so does the absolute amount being eliminated.

Understanding first-order kinetics helps in predicting how long a drug stays in the body and assists in determining appropriate dosing schedules. It helps design medications so they remain effective over a desired period without reaching toxic levels.