# Math Insight

### Overview of: Penicillin clearance model project

#### Project summary

Develop a discrete dynamical system model that describes penicillin clearance by the kidneys, fit the parameter of the model to a dataset, and compare the results of the model to the data.

#### Suggested steps

Here are some suggestions for going about creating and analyzing the mathematical model.

1. Read and possibly rework how we developed and analyzed the model of bacteria growth

2. Read the background on penicillin clearance by the kidneys, focusing on the assumption about the pencillin clearance and how it is similar to the bacteria growth.

3. Introduce appropriate notation to describe the dynamical system you will write. Think about what variable you should use for time. Time in minutes or time measured in a different way?

4. Write a discrete dynamical system model describing the change in penicillin concentration. Compared to the bacteria growth model, you will need a crucial negative sign here. Penicillin is removed at each time step, so the change in penicillin concentration is negative. Since we don't know how fast the kidneys are removing the penicillin, be sure to include a parameter that specifies the speed of penicillin clearance

5. Use the above data to determine the clearance speed parameter from the model. The steps for the bacteria model will work well here. You can use the linear model for population change applet even though we aren't talking about populations here. At this level of description, your penicillin clearance model will have the same form as a population change model. (It may be easier to use if you download the applet and Geogebra on your computer, as described in the applet page.)

Note that it is important that you find one value for the parameter from the entire dataset. Don't determine separate values of the parameter for individual or pairs of data points. According to the word model that forms the basis for this mathematical model, this parameter should stay fixed for all time points.

6. Write a solution to your dynamical system model. Use the value of the parameter determined above and the initial data points as your initial condition. If all goes well, your model should describe exponential decay in the penicillin concentration, so you know exactly the form your solution should take. Assuming that you used a different scale for time than minutes, rewrite your solution to give the penicillin concentration in terms of minutes after bolus injection.

7. Compare the values computed from your solution with the observed data. One way to do this is download the function iteration applet on your computer and run it in Geogebra. (In this applet, you'll have to type in $x$ for the penicillin concentration variable in your dynamical system model.) It's fairly simple to add points corresponding to the data into Geogebra, as detailed with the lead decay model from the dynamical system exploration page. You should find that your model fits the data quite well.

#### Requirements for project report

Your project report should be no longer than three pages. It should contain the following sections.

1. Background: Give a short description of penicillin clearance. Include the role of the kidneys and the word model that you used as a basis of the mathematical model. (No math in this section.)

2. The model: Describe the discrete dynamical system model you used to model the penicillin clearance. Be sure to clearly define all the notation used and identify any parameters. Remember a dynamical system model must always specify initial conditions.

3. Determination of model parameters: Explain how you determined the parameter in your dynamical system model. Either include a printout of a graph used to determine the parameter or sketch what the graph looked like, with clearly labeled axes. Clearly spell out your procedure for determining the parameter from the graph.

4. Model solution: Briefly sketch how you determined the solution to the model. As a service to your reader, you should rewrite the dynamical system after substituting the value of the parameter determined above and using the initial data point as the initial condition. Once you have solved it, rewrite your solution to give the penicillin concentration in terms of minutes after bolus injection.

5. Model comparison: Compare the values computed from your solution with the observed data. Display your results with a graph.

6. Conclusion: Summarize your result. In your summary, evaluate how well the model fit the data in order to conclude how well the word model captured the actual penicillin clearance. Also, mention how close your result was to the result from the literature that 20 percent of the penicillin leaves the blood every five minutes.

Total points: 30