Problem 53 - 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 \(20 \cdot(0.2)^{t}\). (a) The drug has first order elimination kinetics. \(40 \%\) 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 that, 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 \(a_{t+1} = a_t(0.6) + 20 \cdot (0.2)^t\). The maximum drug amount is 20 at \(t = 1\). For large \(t\), the drug amount decreases exponentially.

Step 1: Establish the Recursion Relation

First, we have the word equation \(a_{t+1} = a_{t} + \text{amount absorbed} - \text{amount eliminated}\). We know the amount absorbed from the pill is \(20 \times (0.2)^{t}\), and the drug is eliminated at \(40\%\) per hour, i.e., \(0.4 \times a_t\). Therefore, the recursion relation is given by: \[a_{t+1} = a_t + 20 \cdot (0.2)^t - 0.4 \cdot a_t\] or simplifying, \[a_{t+1} = a_t(0.6) + 20 \cdot (0.2)^t\]

Step 2: Calculate Drug Amount in Blood for \(t = 1\) to \(t = 6\)

Using the recursion relation and knowing \(a_0 = 0\), we calculate iteratively: - For \(t=1\), \(a_1 = 0 \times 0.6 + 20 \times (0.2)^0 = 20\).- For \(t=2\), \(a_2 = 20 \times 0.6 + 20 \times (0.2)^1 = 12 + 4 = 16\).- For \(t=3\), \(a_3 = 16 \times 0.6 + 20 \times (0.2)^2 = 9.6 + 0.8 = 10.4\).- For \(t=4\), \(a_4 = 10.4 \times 0.6 + 20 \times (0.2)^3 = 6.24 + 0.16 = 6.4\).- For \(t=5\), \(a_5 = 6.4 \times 0.6 + 20 \times (0.2)^4 = 3.84 + 0.032 = 3.872\).- For \(t=6\), \(a_6 = 3.872 \times 0.6 + 20 \times (0.2)^5 = 2.3232 + 0.0064 = 2.3296\).

Step 3: Determine the Maximum Drug Amount and When It's Reached

The maximum amount in the interval \(t=1\) to \(t=6\) is calculated by comparing the \(a_t\) values over this period: - The highest amount is at \(t = 1\) with \(a_1 = 20\).Thus, the maximum amount is \(20\) at \(t=1\).

Step 4: Calculate Drug Amount for \(t=0\) to \(t=24\) Using Spreadsheet

Enter the initial condition \(a_0 = 0\) into cell A1 and input the recursion relation formula into the spreadsheet to calculate subsequent values in cells A2 to A25. Adjust the recursive formula as needed to automatically compute drug levels from \(t=0\) to \(t=24\).

Step 5: Show Exponential Decrease with Semilog Plot

Use the calculated values and plot them on semilogarithmic axes (with \(t\) on the x-axis and \(a_t\) on the y-axis in logarithmic scale). As \(t\) becomes large, the curve should approximate a straight line indicating exponential decay of the drug in the blood.

Key Concepts

First Order Elimination Kinetics

First order elimination kinetics is a fundamental concept in pharmacokinetics, describing how drugs are removed from the bloodstream. In this context, first order means that the rate of drug elimination is proportional to the concentration of the drug present in the blood. It's like saying the more drug you have, the faster it will vanish.

For this exercise, 40% of the drug is eliminated every hour, which characterizes it as a first order process. To understand this better, imagine you have a certain amount of drug in your blood, say, 100 units. After one hour, 40% will be gone, leaving you with 60 units. Another hour passes, and another 40% of that remainder is eliminated, leaving you with only 36 units. This exponential reduction is what defines first order kinetics.

• Elimination rate = elimination constant multiplied by drug concentration.
• The elimination constant in our example is 0.4, representing 40% each hour.
• First order processes are common in physiological drug metabolism.

Recursion Relation

A recursion relation provides a way to express quantities at different times in relation to previous times. In drug absorption modeling, recursion relations predict the next concentration level based on the current level, absorption, and elimination.In this exercise, the recursion relation combines these elements:\[ a_{t+1} = a_t(0.6) + 20 \cdot (0.2)^t \]This equation encompasses:

• The current drug amount \(a_t\), reduced by the elimination factor \(0.6 = (1 - 0.4)\).
• The absorption of new drug from the pill, expressed as \(20 \cdot (0.2)^t\).

Recursion is powerful as it builds a timeline of change based on a simple rule. By iteratively applying it, we simulate the dynamic process of drug levels changing over time, capturing the essence of how drugs behave in the body.

Semilogarithmic Plot

A semilogarithmic plot is a graph where one of the axes is on a logarithmic scale, often used in scenarios involving exponential changes. In this exercise, the semilogarithmic plot helps illustrate the exponential decrease of drug concentration in the bloodstream over time. On such a plot:

• The x-axis represents time, in our case, hourly intervals.
• The y-axis, expressed in logarithmic terms, charts drug concentration levels.

By plotting the values on semilogarithmic axes, we easily visualize exponential decay as a straight line—unique to such plots. When we line up our calculations in this format, it validates that as time progresses, the drug’s elimination continues to follow an exponential path. This clear visualization aids in understanding the complex kinetics in a straightforward manner.

Spreadsheet Calculations

Using spreadsheet software to model drug kinetics is effective and illustrative for visualizing temporal changes. The spreadsheet acts as a digital canvas to implement and see the effects of the recursion relation over longer periods.To set this up:

• Start with the initial condition \(a_0 = 0\) in the first cell.
• Input the recursion formula in subsequent cells to calculate \(a_{t+1}\).

As you enter the formula \[ a_{t+1} = a_t(0.6) + 20 \cdot (0.2)^t \]into the spreadsheet, it automatically computes the drug amounts for each hour, from \(t = 0\) to \(t = 24\). This visualizes the concentration over time, showing how it rises initially as the drug is absorbed and then declines due to elimination.Thus, spreadsheets empower students to creatively engage with mathematical concepts, making calculations tangible and easier to explore sequentially through practical visualization.