# Math Insight

### 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

1. 10% of the penicillin in the intestine at the beginning of the period is absorbed into the plasma.
2. 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

1. for 1 g injection,
2. 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 indexMezlocillin concentration (μg/ml)
0071
5156
10245
15333
20425
5 g injection
Time (min)Time indexMezlocillin concentration (μg/ml)
00490
51390
102295
153232
204182