Drug dosing and mathematics

Summary: Mathematicians and scientists calculate optimal drug dosages to help ensure patient health.

Drug dosing is the administration of a particular amount of medication according to a specific schedule. There are two kinds of drugs: prescription drugs and nonprescription drugs (over-the-counter medicine). For prescription drugs, medical doctors normally prescribe the amount and time to take the medication. For over-the-counter drugs, information of drug dosing is usually recommended on the label of the medicine. Drug dosing is common in everyday life, but an error in drug dosing may claim lives or create serious medical burdens. According to a conservative estimate in 2006, drug errors injure more than 2 million Americans per year.

Dosage Measurements

Some drug dosing errors stem from inaccurate measurements and administering improper amounts of chemical compounds to the patient. The first mathematics-related issue is the measurement systems in treatment dosing. Drug dosing normally utilizes the metric system, the apothecary system, or the household system. These are the three main forms of measurement systems in the pharmaceutical industry.

The apothecary system is historically the oldest system in medicine measurement. It consists of grains, drams, ounces, and minims.

Although the apothecary system was widely used during earlier times, it is rarely used in the twenty-first century. The most widespread dosing measurement in liquid drugs in the twenty-first century is the household system, which is rooted on the apothecary system but uses relatively common items as measurement units. The household system primarily consists of teaspoons (tsp), tablespoons (tbsp), ounces (oz), pints (pt), juice glasses, coffee cups, glasses, measuring cups, drops, quarts (qt), and gallons (gal).

The household system is convenient and commonly understandable, but it is just an equivalent measure without specific precision; for instance, the size of a coffee cup may vary. A more scientific and precise way is to measure with the metric system. The metric system is accurate, simple, and popular in most scientific experiments, including drug measurements, even though it is not as handy as the household system. It essentially consists of length, volume, and weight measures.

The basic metric length measure is meter (m). Along with the meter are the following:

The basic metric volume measure is liter (L). Along with the liter are the following:

The basic metric weight measure is gram. Along with the gram are the following:

Each measuring system has its advantages and disadvantages. Administering a drug with a wrong measurement system could result in a fatal error. It is critical to distinguish the different systems and use them appropriately. The following are some basic conversions among the three drug measuring systems.

Dose Response, Drug Dosing, and Statistics

Besides dosage measurement, another important aspect in drug dosing is to understand that because of the immune system and drug resistance, efficacy does not necessarily increase as dosage increases. Factors such as body weight and age affect the shape of the dose response curve for each individual. To take account of population diversity, the expected effect within a population is principally considered as the guideline for the recommended drug dosage. For example, over-the-counter medication normally uses age or body weight of the patient as the guide to recommend efficient dosages.

Similar to the efficacy of a drug, for some medicines, side effects or toxicity of a drug need to be simultaneously considered in drug dosing. If the side effect or toxicity is too strong, administering the medicine may kill (rather than cure) the patient. In this regard, it is necessary to identify the maximum tolerated dose of a drug. The maximum tolerated dose is the largest dosage at which the toxicity/side effect has not reached the level to cause the specific harm to the patient, while the minimum effective dose is the smallest dosage to reach the expected treatment effect of the drug. If the minimum effective dose exceeds the maximum tolerated dose, the drug is normally not permitted. If the minimum effective dose is smaller than the maximum tolerated dose, the dosage range in which the drug is both safe and effective is called the therapeutic window of the drug. For example, if the minimum effective dose of a drug is 5 mg daily and the maximum tolerated dose is 12 mg daily, then the therapeutic window of the drug is 5–12 mg daily.

To make an inference on the efficacy and toxicity of a drug at the same time, statistical methods are used. After clinical trials (such as the double-blind experiment), simultaneous inference methods are used to estimate the minimum effective dose and the maximum tolerated dose. One of the well-known methods in identifying dose effects is Dunnett’s method for multiple comparisons with a control, developed by statistician Charles Dunnett in the mid-twentieth century. Other effective techniques for identifying the therapeutic window of a drug have been explored by mathematicians and statisticians since that time.

Shelf Life

Mathematics and statistics also intertwine with drug dosing on the shelf life of a drug. For medications that emit chemical compounds over time, the drug effect may be affected by chemical half-lives well before the expiration date. In the United States, the Food and Drug Administration (FDA) requires companies to conduct stability analyses to establish the shelf life of new products. The same is true in many other countries around the world. The conclusions are generally based on statistical sampling and mathematical modeling of data, using estimation methods such as simultaneous confidence segments over time.

Bibliography

Boyer, Mary. Math for Nurses: A Pocket Guide to Dosage Calculation and Drug Preparation. 7th ed. Hagerstown, MD: Lippincott Williams & Wilkins, 2008.

Chow, Shein-Chung, and Jun Shao. Statistics in Drug Research: Methodologies and Recent Developments. Boca Raton, FL: CRC Press, 2002.

Lacy C., et al. Drug Information Handbook. 19th ed. Hudson, OH: Lexi-Comp. Inc., 2010.