1. Why Don’t I Feel Better Yet?
SSAC2007.QV4.CC1.3
Why Don’t I Feel Better Yet?
Examining the effect of dose, time interval, and elimination rate on the attainment of therapeutic drug levels
Core Quantitative concept and skill
Forward modeling (what-if analysis)
Orally administered medications may take several doses before their beneficial effects are felt. Why?
Supporting Quantitative concepts and skills
Multivariable function
Visual display and interpretation of data: XY scatter plots
Algebra: Creating equations
Two way tables
Prepared for SSAC by
Cheryl Coolidge - Colby-Sawyer College
© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007

2. Overview of Module
Drug dosing must be carefully planned to ensure that therapeutic levels of the drug are maintained in the patient. Most drugs have a relatively narrow range over which they are safe and effective.
Many factors affect the attainment of therapeutic drugs levels, including the amount of the dose, the time between doses, the absorption and distribution of the drug in the body (determined in large part by its lipid solubility), the way the drug is metabolized, and the way it is eliminated. We will focus only on dose, time interval, and rate of elimination in this module.
Slide 3 presents the problem for you to solve.
Slides introduce you to the use of spreadsheets to perform calculations efficiently.
Slide 7 requires you to create a spreadsheet to examine the effect of dose, dosing interval, and elimination rate.
Slide 8 asks you to create an XY scatter graph of drug level vs. time.
Slide 9 requires you to perform several “what-if” analyses, so that you can understand dosing schedules.
Slides 10 and 11 ask you to study the effect of the elimination rate in detail and teach you how to save the values underlying a formula.
Slide 12 contains the end of module assignments.

3. Problem
You are a physician with a sick, grouchy patient. Your patient has taken one pill of the 40 you prescribed for a 10-day cycle. This patient has called you, wondering why he doesn’t feel better yet. You have to explain the basics of pharmacokinetics to him. You want to review the effect of dose, time of dose, and rate of elimination so that you can be clear when you return his call.
How long does it take to reach the steady-state maximum dose on a dose regimen of 100 mg every six hours, if 40% of the drug has been eliminated from the body when the next dose is given?
In this exercise, we will assume that the absorption and distribution of a drug is instantaneous and that the full dose of the drug is found in the bloodstream immediately after dosing. In reality, of course, absorption and distribution of a drug can take some time, and the dose is diluted by the distribution volume.
We will vary the dose given, the elimination rate, and the time interval of dosing.

4. Skip to Slide 7 if you are comfortable with Excel
Getting Ready: Using a Spreadsheet – Data Input
Skip to Slide 7 if you are comfortable with Excel
Click here to skip ahead
A spreadsheet is an easy way to perform calculations.
The numbers in cells B3 through B7 can just be typed in. As an alternative, Excel can do this for you. Type in the first three values then highlight them (B3 through B5) and place the cursor at the bottom right of the last highlighted cell until you see a small cross. Hold down the left mouse button, drag the pattern through as many cells as you want, and release the button to fill the cells. Excel recognizes the pattern from the first three cells and copies it.
If you want to multiply each of these numbers by 2, a formula can be created to perform this task. In Cell C3, you type the formula as shown. (All formulas begin with =.)
You can copy the formula by clicking on Cell C3 and placing the cursor on the bottom right-hand corner of the cell until you see a small cross.
You then drag the cursor down the column, and your results will be displayed.

5. Getting Ready: Using a Spreadsheet – Calculation Input
Suppose you always want to divide the numbers in Column C by the same number – let’s use 10 for an example. You could create a formula for the first cell in Column C, =C3/10, and drag the formula down Column D as described in Slide 5.
Suppose, though, that you want to divide by a value in a particular cell . So that you don’t have to change the formula for each value in Column C, you can reference the cell (here, C9) in the formula. In your formula, you refer to this cell as an “absolute” (or “fixed”) cell whose position doesn’t change when you copy the formula. To indicate that this cell is absolute, precede both the column and the row number with a dollar sign.
You can make a graph by highlighting a range of data (here, from B3 to C7) and then clicking on the chart wizard button:
You select a graph type (in this case, an XY-scatter plot connected by smooth lines) and follow the directions. Voila! A graph!
When the formula in Cell D3 is copied, the cell referenced in the numerator of the formula will adjust row by row, but the cell referenced in the denominator remains constant.

6. Getting Ready: Using a Spreadsheet – Number Formatting
Depending on the default settings of the version of Excel you are using, the values generated by your equations may display an unnecessary number of decimal places.
To fix this, right-click on the cell or group of cells you wish to change and choose “Format Cells” from the pop-up menu. Select the “Number” tab, and choose “Number” from the “Category” list, if not already selected. In the “Decimal places” scroll box that appears on the right, type in the number of decimal places you would like to use.
When working with percentages in Excel, it is best to treat them as decimals rather than values greater than 1 (e.g., 0.51 instead of 51%). Multiplying your decimals by 100 to obtain percents can, at times, needlessly complicate your equations and hinder Excel’s ability to understand what you’re trying to calculate.
To tell Excel to display the result as a percent, simply highlight the cells with your decimals, and follow the formatting directions previously discussed. However, instead of choosing “Number” from the “Category” list, choose “Percentage”, and select the number of additional decimal places you wish to use.

7. Building the Model Recreate this spreadsheet.
Let’s look at what happens when you give repeated doses of a drug over time.
= cell with a number in it
= cell with a formula in it
Start this spreadsheet by entering the labels and numbers (yellow cells) in the indicated cells. DO NOT enter the values in the cells which are color coded as formulas. In this case, we are assuming that the time interval between doses is 360 minutes (6 hours), the dose is 100 mg, and 40% of the drug has been eliminated from the body when the next dose is given.
Recreate this spreadsheet.
Hint: Enter the elimination rate in Cell C5 as a decimal (0.4). Then format it as a percentage with 0 decimal places. See Slide 6.
In this model, time is the independent variable, and is shown on the x-axis in a graph. The resulting drug level is the dependent variable, which is plotted on the y-axis. The dosing interval, the drug dose, and the elimination rate are parameters. One distinction between parameters and variables is that variables change over time, while parameters remain constant.
Click here for more information about parameters.
Click here for hints on how to build the formulas in Columns E and F.

8. Recreate this graph. Graphing the results
To get a sense of what happens with repeated dosages, you can graph the spreadsheet you have just created. Highlight Cells E3 through F30, click the graph icon, select an XY scatter graph with the dots connected, and add appropriate titles.
Recreate this graph.

9. What-if analysis
The power of a spreadsheet model is that it allows you to examine how the dependent variable changes with changes in the parameters. You don’t have to recreate any formulas; you just have to change the values in the cells reserved for the parameters. The spreadsheet does all the work!
Modeling questions (record your answers):
Using this dosing schedule, what is the maximum drug level that is obtained? At what minimum level does the drug stabilize? What is the fluctuation – the difference between the maximum and the minimum stable levels?
Suppose that this level is too low to be therapeutic. Try increasing the dose to 200 mg, holding all other parameters constant. What happens?
Although the drug levels rise, a wider fluctuation between maximum and minimum levels is obtained. Return the dose to 100 mg, but change the time interval to 180 minutes. What happens?
In reality, giving a drug more frequently generally means that less would have been eliminated. At a dose of 100 mg and an interval of 180 minutes, let’s assume that only 25% of the drug has been eliminated. Make these changes in the spreadsheet. What effect does this have on the maximum and minimum levels and the fluctuation?

10. Effect of elimination rate
Copy the existing spreadsheet onto a new sheet. Place all cells in the same location as before.
Remove the “elim” label and value in Cells B5 and C5.
Add the decimal elimination rates from 0.1 to 0.9 in Row 2. Format as percents.
You will need to modify your formula in Cell F5 to reflect the fact that the elimination rate has been moved.
Position your cursor on Cell F5. The elimination rates are now found in Row 2. You will want the column to adjust as you copy the formula, but not the row. The elimination rate in the formula should be entered as F$2 so the column will adjust but the row will not.
Copy the formulas in Cells F4 and F5 down through Column F by highlighting both.
Now copy from F4 through F30 into Columns G through J. If you have made the appropriate changes, your spreadsheet should match the one at the left.
Suppose you want to graph the effect of changing the elimination rate through a range of 10% to 90%. Every time you change the value in Cell C5 of the spreadsheet, it automatically recalculates, and you lose the drug levels that resulted from the previous rate. So that you can look at a range of parameter values simultaneously, you can set up a two-way table by modifying your spreadsheet. Here’s how:

11. The graph Recreate this graph.
Graph all five elimination rates by highlighting all the columns. To add the series labels, right-click on the chart, select “source data” from the chart menu, select each individual series (upper tabs), then type in the appropriate heading.
Recreate this graph.

12. End of Module Assignments, Homework
Turn in the answers for the questions on Slide 9.
Print and turn in the graph from Slide 8, the graph from Slide 11, and a copy of your saved spreadsheet.
Summarize the effect of elimination rate. Which condition gave you the highest dose levels? Expand your spreadsheet and give the maximum and minimum levels at this elimination rate – i.e., once the rates have stabilized.
Summarize the effect of dose level – both in terms of maximum and minimum dose and fluctuation. Do a graphical analysis as in Slide 10, varying the dose as 100 mg, 250 mg, 500 mg, and 750 mg. Set the elimination rate to 50% (0.5) and the time dose interval to 360 minutes. Turn in this graph.
How would you amend your model if you assume that absorption and distribution are NOT instantaneous, but rather take 10 minutes. Make this change, setting the dose at 100 mg, the interval at 360 minutes, and the elimination rate at 50%. Graph the resulting spreadsheet as before and turn in the graph.
Try to create a dosing regimen that will stabilize at a maximum drug level of 500 mgs with a fluctuation no greater than 150 mgs.
At a dosing schedule of 100 mgs every six hours with a 50% elimination rate, what would happen if you skipped the second dose? Amend your model to reflect this change. .o

13. HINT: Remember, all formulas start with an equals sign: =
Building the formulas
HINT: Remember, all formulas start with an equals sign: =
The value in Cell E4 is derived by adding an increment of time, minutes, to the initial time of 0 minutes. Build the formula to perform this calculation.
Notice that in Cell F4, the entire dose is assumed to have been absorbed in this instantaneous interval. The formula in Cell F4 is obtained by adding the parameter of absolute dose found in Cell C4 (see slide 5 for information about absolute cells) to the previous blood level found in Cell F3, here, 0.
In Cell E5, the parameter of absolute time interval found in Cell C3 is added to the previous time in Cell E4 ( seconds).
In F5, the parameter of absolute percent of the drug is eliminated. To create this formula, start with the amount of drug found in Cell F4, then subtract the product of that amount times the absolute percent eliminated, found in Cell C5. You will need to enclose this product in parentheses when building your formula.
In cells E6 and F6 and beyond, your formulas built in cells E4 through F5 can be copied through row 30. You can copy the block of 4 formulas by highlighting from Cells E4 through F5.
Return to Slide 7.

14. Parameters vs. variables
In a section on frequently misused words in his book The Writer's Art, James J. Kilpatrick quoted a letter from a correspondent, giving examples to illustrate the correct use of the word parameter:
“W.M. Woods...a mathematician...writes... "...a variable is one of the many things a parameter is not." ... The dependent variable, the speed of the car, depends on the independent variable, the position of the gas pedal.”“[Kilpatrick quoting Woods] "Now...the engineers...change the lever arms of the linkage...the speed of the car...will still depend on the pedal position...but in a...different manner. You have changed a parameter"
Return to Slide 7.