Project: Patient Care and Dosing Information

Patient Care and Dosing Information

Introduction

There are many things to consider when administering medications to patients. You are working for a major pharmaceutical company and they want to develop mathematical models for a new trial drug. You will use your ability to analyze rational expressions and set up and solve applications involving variation to make recommendations for the healthcare industry.

Task

In this project you will play the part of a consultant to a pharmaceutical company, which is working closely with hospitals to insure the best care possible for all patients. Working together with your group, you will analyze data and make calculations to determine how prescription drugs enter the bloodstream and the needs associated with patient care. You will make a chart that will allow medical staff to make recommendations for drug dosing.

Instructions

Solve each problem in order and save your work along the way, as you will create a professional report at the conclusion of the project. If required, round to the hundredths place (two decimal places).

· First problem: Established Drug

· A consultant for the pharmaceutical company analyzed data and found that when a certain drug is administered to a patient by injection into the muscle, the following mathematical equation closely models the vital concentration (C) in milligrams per Liter (mg/L) of the drug in the bloodstream at time , where t is measured in hours since giving the drug:

You need to analyze this model to see how the drug enters and leaves the bloodstream in order to answer several questions from healthcare professionals and compare it to a new trial drug. Since the domain is all real numbers greater than or equal to 0, you want to record how the drug changes each hour. Using the model from above, fill in the table:

Time (t) in hours since drug was administered	Concentration of drug (mg/L)
0
1
2
3
4
5
6
7
8
9
10
11
12

· Draw a graph of the drug concentration in mg/L over the twelve hour period. Then answer the following healthcare professionals’ questions (Note: You may want to analyze additional data points to answer some of these questions):

· What happens to the drug concentration over the 12-hour period? Explain in layman terms—everyday words. Include significant details such as what is the highest concentration of drug that is reached in the patient’s bloodstream, when is the highest concentration reached, and how long does the medicine last in the patient’s system.

· It is recommended that the patient always have at minimum 2 mg/L of the drug in the bloodstream. At what point should the patient be given a second dose of the medicine?

· Second Problem: Trial Drug

· The pharmaceutical company is almost finished with a new trial drug that may replace the established one from Problem 1. This drug will be injected directly into the bloodstream. They have asked their analysts to develop a mathematical model for the vital concentration (C) in milligrams per Liter (mg/L) of the drug in the bloodstream at time , where t is measured in hours since giving the drug. Since it is difficult to model data with rational expressions, your job is to simplify the analysts’ models and compare them to externally collected data to determine which one best fits the data.

Model 1: , Model 2: , Model 3:

· External data was collected on patients by observing and recording the concentration in mg/L of the trial drug. Which of the three simplified models best fits this data?

Time (t) in hours since drug was administered	External Data Concentration of drug (mg/L)	Model 1 Concentration of drug (mg/L)	Model 2 Concentration of drug (mg/L)	Model 3 Concentration of drug (mg/L)
0	28
1	14
2	10
3	7
4	6
5	5
6	4
7	3.5
8	3
9	2.8
10	2.6
11	2.3
12	2.2

· Finally, you need to make an analysis similar to before by drawing a graph of the simplified model that you determined best fits the external data. Then answer the following healthcare professionals’ questions (Note: You may want to analyze additional data points to answer some of these questions):

· It is recommended that the patient always have at minimum 2 mg/L of the drug in the bloodstream. At what point should the patient be given a second dose of the medicine?

· Why would the trial drug be considered an improvement over the previous medicine? Why would the established drug from Problem 1 be considered better than the trial drug? You need to determine which drug (established or trial) your group wants to use in patient care. You will use this model in Problem 4.

· Third Problem:

· One of the problems with the dosing of medicine is that the amount of blood in a human is proportional to its body size. This means that a larger person has more blood and thus, needs a larger dose of medicine. You find that the volume of blood has a direct variation to body weight. The data for different adults are shown in the chart below:

Body Weight in Pounds	Blood Volume in Liters
200	7.2
154	5.5
100	3.6

The direct variation equation would be of the form , where V is the volume of blood in Liters, k is the constant of proportionality, and W is the body weight in pounds. Solve for the constant of proportionality. Check your results with all of the data values to determine if this is truly a direct variation situation.

· Based on your direct variation equation, fill in the following table for each body weight, so that healthcare professionals can easily determine the amount of blood in liters of a patient based on his/her recorded weight.

Body Weight in Pounds	Blood Volume in Liters
100
110
120
130
140
150
160
170
180
190
200

· Fourth Problem:

· In order to determine the critical amount of medicine in milligrams that is needed for a patient at a given time, use the table from the Established Drug in Problem 1 or the Trial Drug in Problem 2 (whichever one your group selected in Problem 2) and multiply by the amount of liters of blood for a 150 and 180 pound person. This calculation will determine the number of milligrams of medicine that is needed for a 150 and a 180 pound person to reach the vital concentration. Then calculate the difference in milligrams.

Time (t) in hours since drug was administered	Concentration of drug (mg/L) (Established or Trial)	150 pound Amount of drug (mg)	180 pound Amount of drug (mg)	Difference per 30 pounds (mg)
0
1
2
3
4
5
6
7
8
9
10
11
12

· You need to determine how much additional medicine should be given to patients with different weight requirements. Since the direct variation equation is linear, the calculated difference will be the same for each additional 30 pounds. If you consider the difference in milligrams in the first five hours after the medicine is administered, what recommendation would you make? Your statement may be something such as, “For each additional 30 pounds above 150 pounds, I would increase the medicine by _____________ milligrams”.

Collaboration

Get together with another group to compare your answers to each of the four problems. Discuss how you might combine your answers to make a more complete and convincing analysis of the situation.

Conclusions

Present your solution in a way that makes it easy for healthcare professionals, such as doctors and nurses, to understand your results. Be sure to clearly explain your reasoning at each stage and conclude with a recommendation to either pursue the trial drug or stick with the established drug. Regardless which drug your group recommends, be sure to explain why the drug is more beneficial to patients. Finally, explain how patients’ weights will have an impact on the critical concentration of the drug in the bloodstream.

Grade

Your project will be given a score of 1 to 4, with 4 being the highest score possible. Your project should correctly identify and justify mathematically: (1) graphs of the models for the established and trial drug with explanations in layman terms—everyday words, (2) simplified models for the trial drug with the correct model identified, (3) the direct variation equation with a chart of the number of liters by weight, and (4) the additional medication requirements based on weight. You will be evaluated based on the following:

Score

Content

Presentation/Communication

• The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution.

• The solution completely addresses all mathematical components presented in the task.

• The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results.

• Mathematically relevant observations and/or connections are made.

• There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made.

• Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem.

• There is precise and appropriate use of mathematical terminology and notation.

• Your project is professional looking with graphics and effective use of color.

• The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution.

• The solution addresses all of the mathematical components presented in the task.

• The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem.

• Most parts of the project are correct with only minor mathematical errors.

• There is a clear explanation.

• There is appropriate use of accurate mathematical representation.

• There is effective use of mathematical terminology and notation.

• Your project is neat with graphics and effective use of color.

• The solution is not complete indicating that parts of the problem are not understood.

• The solution addresses some, but not all of the mathematical components presented in the task.

• The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning.

• Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures.

• Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics.

• There is some use of appropriate mathematical representation.

• There is some use of mathematical terminology and notation appropriate for the problem.

• Your project contains low quality graphics and colors that do not add interest to the project.

• There is no solution, or the solution has no relationship to the task.

• No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem.

• The solution addresses none of the mathematical components presented in the task.

• There were so many errors in mathematical procedures that the problem could not be solved.

• There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem.

• There is no use, or inappropriate use, of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.).

• There is no use, or mostly inappropriate use, of mathematical terminology and notation.

• Your project is missing graphics and uses little to no color.

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Monterey Institute for Technology and Education 2011