### Penicillin clearance model exercises

The penicillin clearance model page explained how to develop a discrete dynamical system model based on experimental measurements of penicillin in the blood. In the following exercises, you can explore similar models and datasets.

#### Exercise 1

A one-liter flask contains one liter of distilled water and 2 g of salt. Repeatedly, 50 ml of solution are removed from the flask and discarded after which 50 ml of distilled water are added to the flask. Introduce notation and write a dynamic equation that will describe the change of salt in the beaker each cycle of removal and replacement. How much salt is in the beaker after 20 cycles of removal?

#### Exercise 2

A 500 milligram penicillin pill is swallowed and immediately enters the intestine. Every five minute period after ingestion of the pill

- 10% of the penicillin in the intestine at the beginning of the period is absorbed into the plasma.
- 15% of the penicillin in the plasma at the beginning of the period is removed by the kidney.

Let $I_t$ be the amount of penicillin in the intestine and $S_t$ be the amount of penicillin in the plasma at the end of the $t^{th}$ five minute period after ingestion of the pill. Complete the following equations, including + and - signs.

Initial conditions | |

$I_0 =$ | |

$S_0 =$ |

Penicillin change per time period | Penicillin removed | Penicillin added | |

${I_{t+1} \nobreak{-} I_t \nobreak{=}}$ | |||

${S_{t+1} \nobreak{-} S_t \nobreak{=}}$ |

#### Exercise 3

Along with the data for 2 g bolus injection of mezlocillin, T. Bergan reported serum mezlocillin concentrations following 1 g bolus injection in healthy volunteers and also data following 5 g injection in healthy volunteers. Data for the first twenty minutes of each experiment are shown below.

Following the step given in the penicillin clearance model page, analyze the data

- for 1 g injection,
- for 5 g injection.

Prepare a table and graph to compare your computed solution with the observed data.

Plasma mezlocillin concentrations at five minute intervals following
injection of either 1 g of mezlocillin or 5 g of mezlocillin into
healthy volunteers. Taken from T. Bergans, Penicillins, in *Antibiotics and Chemotherapy*, Vol 25, H. Schøonfeld, Ed., S. Karger, Basel, New York, 1978.

1 g injection | ||
---|---|---|

Time (min) | Time index | Mezlocillin concentration (μg/ml) |

0 | 0 | 71 |

5 | 1 | 56 |

10 | 2 | 45 |

15 | 3 | 33 |

20 | 4 | 25 |

5 g injection | ||
---|---|---|

Time (min) | Time index | Mezlocillin concentration (μg/ml) |

0 | 0 | 490 |

5 | 1 | 390 |

10 | 2 | 295 |

15 | 3 | 232 |

20 | 4 | 182 |

#### Selected answers

Once you've worked out some of these exercises, you can check your work with the answers to selected problems.

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