One Compartment Drug


Moussa Ndiaye

Unit Title:

One Compartment Drug





Estimated Duration:

10-53 min

Unit Activities:

Activity 1: Introduction

Activity 2: Investigating Pharmacokinetics Data 

Activity 3: Analyzing Pharmacokinetics Data

Activity 4: Creating a prospective Model 

Activity 5: Presentation 

Background Knowledge: 

Now would be a good time to make sure that your students have been introduced to pharmacokinetics.  The following is a list of concepts that students should be familiar with in order to be successful:

The LADME scheme

Drug distribution

Drug elimination

Pharmacokinetic models

Therapeutic drug monitoring

Website link

Website link

Know how to use the FDA website 

Know how to extract pharmacokinetics  from a drug label

Rate of change

Exponential and Logarithmic functions 


July 2014

The Big Idea (including global relevance)

"Laughter is the best medicine"

There is always the quest to not only find new drugs but also to find or develop new ways of administering the medication. The goal of all drug delivery systems is to place medication intact and to the specific part of the body.  This is done through a device that can control the amount administered and the duration of the administration either by a physiological or by a chemical trigger or by mechanical tools. 

The Essential Question

How do medical practitioners administer drugs?

What could explain why some people use drugs and others do not?

How are drugs distributed in the body?

Justification for Selection of Content

This is the first time we have a group of 10th graders taking AP calculus at Shroder HS. They must have a deep understanding of derivatives and limits in order for them to have any chances of passing the AP exam that will be administered at the end of school year. Going from slope as rate of change to instantaneous rate of change to maximization to surface area (integral) to sequence and series, will help students have a better understanding of calculus. After this year we will rely on the AP result to improve or to adjust this unit.

The Challenge

Design a schedule for short-term or long-term administration of the drug that will:

  • Get the concentration to an effective level as soon as possible
  • Keep the concentration below a known safe maximum level CS, and
  • Keep the concentration above a known effective level CE.
The Hook

Have a Doctor or a Pharmacist as guest speaker

Teacher's Guiding Questions
  1. How does the drug enter the body?
  2. How does it disperse throughout the body?
  3. How is the drug broken down by the body?
  4. How is the drug eliminated from the body?
  5. How are these factors important?
  6. How long will the drug stay in a person’s system/body?
  7. When is the medication harmful for the body?
  8. Is there a safety factor/limit to be considered?
  9. What is the rate at which the effectiveness reduces?
ACS (Real world applications; career connections; societal impact)

Real world applications:

The safe and effective administration of medications occurs in many areas including hospitals and medical facilities as well as in our homes.  The ability to administer correct dosages is an important part of ever y day life, particularly for older people.


Societal impact:

  • May prolong life of those living with chronic disease
  • Provide pain relief
  • Improve quality of life
  • Improve body vital body functions
Engineering Design Process (EDP)

1. Problem is to design a schedule for administration of the drug that will get the concentration level to an effective  and safe level

2. Identify Criteria & Constraints

  • Drugs must be administered intravenously  (single Rapid injection , IV infusion, single rapid Injection Followed by IV infusion only)
  • Research and investigate. Each group will research at least 10 drugs (at least 2 drugs per student)
  • Use Internet FDA  website

3. Brainstorm Possible Solutions

  • Student will brainstorm in group about what makes a schedule easy to understand easy to use in Administrating drug

4. & 5. Generate Ideas/Explore Possibilities

  • Each group will develop two or three ideas  that a layperson will be able to understand
  • Discuss developed ideas with team members

6. Select an Approach

  • Now it is time to select the best solution for further development ( Pro/Cons of each design) and Why the team selected the final design

7. Build a Model or Prototype of the Solution

8. Revise and Refine the Solution

Use peer feedback, to make sure your design meets all the constraints:

Drug schedule that will:

  • Get the concentration to an effective level as soon as possible
  • Keep the concentration below a known safe maximum level
  • Keep the concentration above a known effective level
Unit Academic Standard

The College Board state that:

  • Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation, and should be able to use derivatives to solve a variety of problems.
  • Students should be able to communicate mathematics and explain solutions to problems both verbally and in written sentences.
  • Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral.
  • Students should be able to use technology to help solve problems, experiment, interpret results, and support conclusions.
  • Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
Where the CBL and EDP appear in the Unit

Lesson 1: investigation

  • Activity 2:  Pharmacokinetics Data (EDP)
  • Activity 3: Analyzing Pharmacokinetics Data (CBL, EDP)

Lesson 2

  • Activity 4: Modeling (CBL, EDP)

The mathematics behind medicinal drug administration is more than how many pills or what is the dose of the medication.

Additional Resources

Here are 5 YouTube video that will help put pharmacokinetics into perspective

Pre-Unit Assessment Instrument
Post-Unit Assessment Instrument
How to Make This a Hierarchical Unit

Determining the Relationship of Insulin to Diabetes

The 7th Grade math standards focus on such things as:

  • ratios and proportions
  • solving real-life and mathematical problems using numerical and algebraic expressions and equations

The 8th grade math standards focus on such things as:

  • modeling an association in bivariate data with a linear equation
  • solving linear equations
  • understanding the Pythagorean Theorem

This unit could be changed to investigate linear relationships in medicine such as the relationship between blood sugar level and time (  Students might still prepare a brochure or presentation to illustrate a drug regimen that would be effective in controlling a disease such as diabetes.


One of my biggest concerns was that the subject matter of Pharmacokinetics would be challenging for my 10th graders.  Covering a large amount of material in my AP Calculus block is already very difficult for this group of students, but luckily I have the students for two consecutive periods which in small part removes some of the obstacles of having to find time to cover all of my planned lessons.  So I was able to do the unit in 5 lessons and it was very successful.  My students showed genuine interest in the units and were excited to discuss the different possibilities of more efficient ways to deliver medicine to patients.

Dr Giovanni Pauletti Director of the graduate program in Pharmaceutical Sciences was instrumental in getting my students getting and holding the students’ interest with his introductory lesson introducing the concepts in the unit.  He was recommended by the dean of the College of Pharmacy at the University of Cincinnati. The best way to make the material accessible to students is to get the assistance from the professionals who are working in the field. They know how to best present the work that they do so that students are interested.  The proof of the effectiveness of Dr. Pauletti is that the students were talking about pharmacokinetics with other teachers.  What a triumph!

Covering this unit was successful because the students covered math problems in the context of pharmacokinetics. 

Students used equations in a context to find the minimum and maximum concentration.

Students used scenarios to communicate their understanding of medications. 

The next time I will divide this units into 3 units because it will be more beneficial for student’s to start with basic drug administration to give them a more comprehensive approach to pharmacokinetics. My divisions will be as followed:

Unit 1:  Oral Administration vs Rapid Infusion; Rate of Change

Unit 2: One Compartment Drug; Area Under Curve (AUC)

Unit 3: Two Compartment Drug; Second Order of Differential Equations

What I would do differently:

  • Break the project into manageable sections
  • Set concrete deadlines for each part of the project in order to have student be responsible for turning in completed work.
  • Do a better job at clearly defining requirements for each part of the project
  • When taking videos of student presentations,  I would use a camera with a tripod so that you do not get “jittery” videos
  • During the unit, I will make sure that students understand where they are at in the engineering design process.