1.6: Constructing a Mathematical Model of Penicillin Clearance.
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In this section you will build a model of depletion of penicillin from plasma in a patient who has received a bolus injection of penicillin into the plasma.
When penicillin was first discovered, its usefulness was limited by the efficiency with which the kidney eliminates penicillin from the blood plasma passing through it. ”Penicillin is actively excreted, and about 80% of a penicillin dose is cleared from the body within three to four hours of administration. Indeed, during the early penicillin era, the drug was so scarce and so highly valued that it became common to collect the urine from patients being treated, so that the penicillin in the urine could be isolated and reused.11” The modifications that have been made to penicillin (leading to amphicillin, moxicillin, mezlocillin, etc.) have enhanced its ability to cross membranes and reach targeted infections and reduced the rate at which the kidney clears the plasma of penicillin. Even with these improvements in penicillin, the kidneys remove 20 percent of the penicillin in the plasma passing through them. Furthermore, all of the blood plasma of a human passes through the kidneys in about 5 minutes. We therefore formulate
Mathematical Model 1.6.1 Renal clearance of penicillin. In each five minute interval following penicillin injection, the kidneys remove a fixed fraction of the penicillin that was in the plasma at the beginning of the five minute interval.
We believe the fraction to be about 0.2 so that 5 minutes after injection of penicillin into a vein, only 80% of the penicillin remains. This seems surprising and should cause you to question our assertion about renal clearance of penicillin12 . See Also: mathinsight.org/penicillin_clearance_model
An amount of drug placed instantaneously (that is, over a short period of time) in a body compartment (plasma, stomach, muscle, etc.) is referred to as a bolus injection. An alternative procedure is continuous infusion as occurs with an intravenous application of a drug. Geologists refer to a meteor entering the atmosphere as a bolide. Data for penicillin concentration following a 2 g bolus injection are shown in Figure 1.6.1.
Explore 1.6.1 We think you will be able to develop this model without reading our development, and will find it much more interesting than reading a solution. You have a mathematical model and relevant data. You need to
Step 2. Introduce appropriate notation.
Step 3. Write a dynamic equation describing the change in penicillin concentration. You will need a crucial negative sign here, because penicillin is removed, and the change in penicillin concentration is negative. Evaluate a parameter of the dynamic equation.
Figure \(\PageIndex{1}\): Data for serum penicillin concentrations at 5 minute intervals for the 20 minutes following a ‘bolus’ injection of 2 g into the serum of “five healthy young volunteers (read ‘medical students’)” taken from T. Bergans, Penicillins, in Antibiotics and Chemotherapy, Vol 25, H. Schøonfeld, Ed., S. Karger, Basel, New York, 1978. We are interpreting serum in this case to be plasma.
Step 4. Review the mathematical model (it will be excellent and this step can be skipped).
Step 5. Write a solution equation to the dynamic equation.
Step 6. Compare values computed with the solution equation with the observed data (you will find a very good fit).
You will find that from this data, 23 percent of the mezlocillin leaves the serum every five minutes.
Exercises for Section 1.6, Constructing a Mathematical Model of Penicillin Clearance.
Exercise 1.6.1 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 1.6.2 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.
Exercise 1.6.3 Along with the data for 2 gm 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 in Table Ex. 1.6.3.
- Analyze the data for 1 g injection as prescribed below.
- Analyze the data for 5 g injection as prescribed below.
Analysis. Compute \(P_{t+1} − P_{t}\) for \(t = 0, 1, 2, 3\) and find a straight line passing through (0,0), (\(y = mx\)), close to the graph of \(P_{t+1} − P_{t} vs P_{t}\) . Compute a solution to \(P_{t+1} − P_{t} = m P_{t}\), and use your solution equation to compute estimated values of \(P_t\). Prepare a table and graph to compare your computed solution with the observed data.
| 1 g injection | 5 g injection | ||||
|---|---|---|---|---|---|
| Time (min) | Time index | Mezlocillin concentration \(\mu g / ml\) | Time (min) | Time index | Mezlocillin concentration \(\mu g / ml\) |
| 0 | 0 | 71 | 0 | 0 | 490 |
| 5 | 1 | 56 | 5 | 1 | 390 |
| 10 | 2 | 45 | 10 | 2 | 295 |
| 15 | 3 | 33 | 15 | 3 | 232 |
| 20 | 4 | 25 | 20 | 4 | 182 |
Exercise 1.6.4 Two liters of fresh water (distilled H2O) is contained in a plastic bag that is floating in the ocean. A small area of a side of the plastic bag is a semipermeable membrane (permeable to water but not to salt). Osmosis will drive the water out of the plastic bag and into the ocean.
Write a mathematical model of the transfer of water from the plastic bag to the ocean. Introduce notation and write an initial condition and a difference equation with that will describe the amount of water in the small container.
Exercise 1.6.5 There is a standard osmosis experiment in biology laboratory as follows.
Material: A thistle tube, a 1 liter flask, some ‘salt water’, and some pure water, a membrane that is impermeable to the salt and is permeable to the water.
The bulb of the thistle tube is filled with salt water, the membrane is placed across the open part of the bulb, and the bulb is inverted in a flask of pure water so that the top of the pure water is at the juncture of the bulb with the stem.
Because of osmotic pressure the pure water will cross the membrane pushing water up the stem of the thistle tube until the increase in pressure inside the bulb due to the water in the stem matches the osmotic pressure across the membrane.
Our problem is to describe the height of the water in the stem as a function of time. The following mathematical model would be appropriate.
Mathematical Model. The amount of water crossing the membrane during any minute is proportional to the osmotic pressure across the membrane minus the pressure due to the water in the stem at the beginning of the minute.
Assume that the volume of the bulb is much larger than the volume of the stem so that the concentration of ‘salt’ in the thistle tube may be assumed to be constant, thus making the osmotic pressure a constant, \(P_0\) (This is fortunate!). Assume that the radius of the thistle tube stem is \(r\) cm. Then an amount, \(V\) cm3, of additional water inside the thistle tube will cause the water to rise \(V /\left(\pi r^{2}\right)\) cm. Assume the density of the salt water to be \(\delta\) gm/cm.
Introduce notation and write a difference equation with initial condition that will describe the height of the water in the stem as a function of time.
11 http://en.Wikipedia.org/wiki/Penicillin, which cites Silverthorn, D.U., Human Physiology: An Integrated Approach (3rd Ed.) Upper Saddle River, NJ, Pearson Education
12 That all of the plasma passes through the kidney in 5 minutes is taken from Rodney A. Rhoades and George A. Tanner, Medical Physiology, Little, Brown and Company, Boston, 1995. “In resting, healthy, young adult men, renal blood flow averages about 1.2 L/min”, page 426, and “The blood volume is normally 5-6 L in men and 4.5-5.5 L in women.”, page 210. “Hematocrit values of the blood of health adults are \(47 \pm 5%\) for men and \(42 \pm 5%\) for women”, page 210 suggests that the amount of plasma in a male is about \(6 L \times 0.53 = 3.18 L\). J. A. Webber and W. J. Wheeler, Antimicrobial and pharmacokinetic properties, in Chemistry and Biology of \(\beta\)-Lactin Antibiotics, Vol. 1, Robert B. Morin and Marvin Gorman, Eds. Academic Press, New York, 1982, page 408 report plasma renal clearances of penicillin ranging from 79 to 273 ml/min. Plasma (blood minus blood cells) is approximately 53% of the blood so plasma flow through the kidney is about \(6 liters \times 0.53/5 min = 0.636 l/min\). Clearance of 20% of the plasma yields plasma penicillin clearance of \(0.636 = 0.2 = 0.127 l/min = 127 ml/min\) which is between 79 and 273 ml/min.