SYSTEMS OF MEASURE

Summary of conversion between metric, apothecaries’ and avoirdupois systems:

Note that in the apothecaries’ and avoirdupois systems there is only one common unit of measure, the grain. The other measurement units carry different values when comparing the systems. When converting between the two, the pharmacist should convert the value down to the grain amount in the one system, then convert to the other system.

Per the United States Pharmacopeia, 1 grain = 64.8 mg.

AVOIRDUPOIS SYSTEM

A system of masses based on a pound weighing 16 ounces mostly commonly used in the United States for commercial purposes.

RATIO AND PROPORTIONS

RATIO

A ratio is a comparison of two numbers. In pharmacy calculations, a ratio can be used to express strength of drug concentration.

Example: A 1:25 solution of wintergreen oil means that 1 mL of wintergreen oil is contained in each 25 mL of solution.

PROPORTIONS

A proportion represents the equality between two ratios. A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. This mathematical concept is often used in community pharmacy.

Example

If 5 tablets contain 1625 mg of acetaminophen, how many tablets should contain 2925 mg?

Relevant measurements and conversions

DIMENSIONAL ANALYSIS

Dimensional analysis is a method of manipulating units to solve mathematical equations. The process allows you to cancel out unwanted units leaving only those units you want your answer to be expressed as.

Example

A pharmacist wants to know how many inhalers should be dispensed to a patient to provide a 60-day supply of fluticasone. The recommended daily dose is 250 mcg twice daily. The commercial inhaler delivers 220 mcg per metered dose and contains 60 metered inhalations.

Solution

The pharmacist should dispense 2 inhalers for a 60-day supply.

INTERPRETATION OF MEDICATION ORDERS

PERCENTAGE AND RATIO STRENGTH CALCULATIONS EXPRESSED AS V/V, W/W, AND W/V

Certain prescriptions are expressed in weight/weight (w/w), volume/volume (v/v), and weight/volume (w/v) percentages. To properly process prescription orders of this nature, the pharmacist must be able to make conversions and calculations with these units.

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V/V: If the solute and the preparation are expressed in the same units, then concentration is dimensionless. For example, 10 mL of alcohol dissolved in a sufficient quantity of water to make 40 mL of solution is dimensionless: 10 mL/40 mL= 0.250 (or 25% v/v).

W/W: If the quantity of solute and of the preparation are expressed in the same units of weight, the concentration is dimensionless. If 10 g of charcoal are mixed with 65 g of another powder to make a total of 75 g, the charcoal concentration is 10 g/75 g = 0.133 by weight (or 13.3% w/w).

W/V: When a solute is measured by weight and the solution by volume, concentration is not dimensionless. If 1.25 g of NaCl is dissolved in sufficient water to make 55 mL of solution, the concentration is 1.25 g/55 mL = 0.0227 g/mL (w/v). The % w/v is expressed as #g/100 mL (e.g., 2.27% or 2.27 g/100 mL).

Example 1:

How many grams of drug should be used to prepare 120 grams of a 2% w/w solution in water?

Example 2

What is the percentage strength (w/v) of a solution of drug if 40 mL contain 5 grams?

PPM AND PPB (PARTS PER MILLION AND PARTS PER BILLION)

When ppm or ppb is used as a designation for concentration, some systems are w/w, some are v/v and some are w/v. Concentration is always a ratio or fraction in w/w and v/v situations. Weight by volume (w/v) concentrations are always defined in terms of grams and milliliters. The same default rules are followed as for percentage systems.

Example

Express 2 ppm of ferrous gluconate in water in percentage strength and ratio strength.

DILUTION, CONCENTRATION, AND ALLIGATION

DILUTION OF AN INGREDIENT

Dilution is the addition of diluent to the ingredient or an admixture of the ingredient with solutions to achieve a lower concentration of solution.

Example

A 1:5000 dilution of drug A is requested. If 1 mL of drug A injection 1:200 is mixed with sterile water for injection, how many mL of water will be needed?

CONCENTRATION OF AN INGREDIENT

Concentration is the addition of an active ingredient or evaporation of the diluent from an active ingredient to create a more concentrated solution.

Example

How many grams of coal tar containing 25% (w/w) should be added to petrolatum to prepare 240 grams of coal tar containing 15% (w/w)?

Solution

The coal tar (active ingredient) is added to a diluent (petroleum) currently containing no coal tar.

ALIQUOT METHOD (ALLIGATION)

Alligation is a method that is particularly useful when mixing two or more preparations of known strengths to prepare a mixture of an intermediate desired strength. The final mixture will be an average of the individual strengths, which are calculated as proportional parts.

Alligation Alternate and Alligation Medial are methods that can be used to solve any type of dilution or concentration problem, including concentrations expressed in mg/mL, ratios, mixtures of liquids of known specific gravities, etc. The strengths of all preparations being mixed and the final mixture are expressed in a common denomination (of weight, volume, percentage, etc.) when setting up the alligation equation. When diluting a preparation, the strength of the diluent is considered to be 0%. When increasing the strength of a given mixture by adding more drug/active ingredient, the strength of the active ingredient to be added is considered to be 100%. A final proportion allows a correlation between the parts and any specific denomination needed.

Example 1

A pharmacist has a 60% solution and a 15% solution. She needs a 40% solution to compound a medication. What is the proportion of the 60% and 15% solutions that would make a 40% solution? This example will use the process of Alligation Alternate to calculate the quantities of each mixture needed to make the final mixture of the desired strength:

Solution

25 parts of the 60% solution combined with 25 parts of 15% solution would yield 45 parts of a 40% solution.

ISOTONIC SOLUTIONS

Osmosis occurs when a solvent (e.g.,water) passes through a semipermiable membrane from a low-concentration solution into a high-concentration one, with the result that the concentrations become equalized. The pressure that causes this occurrence is known as osmotic pressure.

A solution that exerts the same osmotic pressure as a specific body fluid is known as isotonic. If the solution exerts an osmotic pressure lower than that of specific body fluid, the solution is hypotonic. If the actual solution exerts an osmotic pressure higher than that of specific body fluid, the solution is considered hypertonic.

Example

How much sodium chloride is needed to adjust the following prescription to isotonicity? (E value [sodium chloride equivalents] for zinc sulfate is 0.15)

Zinc sulfate 2%

NaCl q.s.

Purified water q.s. 60 mL

Make isotonic solution

ELECTROLYTE SOLUTIONS

Electrolyte solutions are used to treat fluid and electrolyte disturbances. They may be prepared as oral solutions, syrups, dry granules intended to be dissolved in water or juice to make an oral solution, or oral tablets or capsules, and they are also commonly prepared as intravenous infusions.

To convert electrolytes in solution (expressed as milliequivalents [mEq] per unit volume to weight per unit volume or vice versa), the following calculation may be used:

Table 2-1 Valences and Atomic Weights of Select Ions

Example

How much calcium chloride (CaCl2•2H₂O) is required to prepare 100, 1 mL ampules containing 10 mEq per mL? (mw = 147)

TPN CALCULATIONS

Total parenteral nutrition (TPN) provides all of the patient’s daily nutritional requirements and generally contains dextrose (carbohydrate), amino acids (protein source), vitamins, trace minerals, electrolytes, and fat emulsions. TPN solutions may also include insulin and occasionally therapeutic drugs. The amount of protein, dextrose, and fat are calculated based on the patient’s daily kcal (calories) needed and available stock solutions. Other ingredients do not contain calories.

Example

A patient needs 1600 kcal/day. The physician has ordered that the patient receive 65% of the daily calories (kcal) from carbohydrates, 10% from protein, and 25% from fat.

Calculate the amount (volume) needed to supply the dextrose, protein, and fat calories from these pharmacy stock solutions:

Dextrose 65%, amino acid 10%, fat 25%

First, determine how many kcal the patient needs from each component:

Next, convert these kcals into grams:

Then, calculate how many milliliters are needed from each stock solution:

NOTE:

Carbohydrate contains 3.4 kcal/g

Amino acid contains 4 kcal/g

Fat contains 9 kcal/g

CALCULATION OF DOSES

There are a variety of ways to determine doses of drugs including by age, body weight, surface area, creatinine clearance, and other pharmacokinetic parameters.

CREATININE CLEARANCE

When using the below equations, two factors to consider are (1) the serum creatinine is at steady state and (2) the weight, gender, and age of the individual reflect normal muscle mass.

Cockcroft-Gault equation

To estimate renal function for the purpose of drug dosing, creatinine clearance should be measured or estimated.

If the individual is obese or not within 30% of their ideal body weight, other methods of calculating creatinine clearance should be used. Ideal body weight (IBW) or adjusted body weight (ideal body weight plus 40% of obese weight) instead of actual body weight in the Cockcroft-Gault equation will provide a better estimate of creatinine clearance.

BODY SURFACE AREA

The practioner may need to take into account body surface area as a possible variable when determining drug dosage (e.g., chemotherapy).

Body Surface Area (BSA) – The Mosteller Formula:

The average BSA of an adult is 1.73 m².

CALCULATIONS FOR PEDIATRIC DOSES

Various pediatric formulas have been used historically to calculate APPROXIMATE pediatric dosages.

Young’s rule, based on age:

Fried’s rule for infants:

Clark’s rule, based on weight:

BSA approximation of child’s dose:

STOCK SOLUTIONS

A stock solution, commonly referred to as bulk bottle, is a large volume of a reagent (in chemistry) or in this case, medication. These stock solutions can be prepared by a manufacturer or compounded in the pharmacy. Pharmacists typically take stock solutions and use them to prepare weaker solutions of medications or chemicals for laboratory or clinical use.

Example

How many mL of a 0.5% gentian violet stock solution is needed to prepare 1 pint of a 1:2000 solution?

RECONSTITUTION OF DRY POWDERS

Many drugs (antibiotics, steroids, and biologics) that are not stable in solution are prepared as dry-filled solids or lyophilized powders. Prior to use, these dry powders must be reconstituted as a solution with a suitable diluent in the proper volume to give specified concentration (usually provided in the package insert). Occasionally, the physician may prescribe a final concentration different from the one provided by the manufacturer. Also, in some cases, the pharmacist needs to determine if the powdered drug contributes to the final volume of the reconstituted solution before modifying the label instructions.

Example

The package information of a vial containing 30 million units of penicillin G potassium specifies that when the appropriate amount of sterile solvent is added to dry powder, the resulting concentration is 500,000 units per mL. How many milliliters of sterile water for injection are needed to prepare the following solution? (Note: the powder accounts for 8 mL of the final volume)

Penicillin G potassium 30,000,000 units

Sterile water for injection

Provide a solution containing 500,000 units per mL

INTRAVENOUS INFUSIONS, PARENTERAL ADMIXTURES, AND FLOW RATES

Intravenous infusions are large volumes of sterile, aqueous preparations administered intravenously (through a vein) over an extended period of time.

Example

An order is written for 25,000 units of heparin in 250 mL of D5W to infuse at 2000 units/hr. What is the correct rate of the infusion (in mL/hr)?

Solution

Parenteral admixtures are a sterile preparation that involves the combination of one or more drugs to large-volume.

Example

A patient weighs 170 pounds. A pharmacist receives a prescription order for 0.25 mg amphotericin B per kilogram body weight. How many milliliters of a 25 mg/10 mL solution are needed to supply the dose, which will then be diluted in 500 mL of 5% dextrose?

Solution

Calculating IV flow or drip rates are necessary to ensure that the patient is receiving the desired amount of drug that was ordered.

Example

If 20 mg of drug is added to a 750 mL parenteral fluid, what flow rate, in millilters per hour, will deliver 2 mg of drug per hour?

Solution

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kilo-	10³	1 thousand (1000) times the basic unit
hecto-	10²	1 hundred (100) times the basic unit
deka-	10	1 ten (10) times the basic unit
deci-	10⁻¹	1 tenth (0.1) times the basic unit
centi-	10⁻²	1 hundredth (0.01) times the basic unit
milli-	10⁻³	1 thousandth (0.001) times the basic unit
micro-	10⁻⁶	1 millionth times the basic unit
nano-	10⁻⁹	1 billionth times the basic unit
pico-	10⁻¹²	1 trillionth times the basic unit

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Pharmaceutical Calculations

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