CHAPTER 2 Pharmaceutical Calculations
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.
|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|
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.
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.
Certain medications that are administered or dispensed to a patient come in the form of liquids and are administered as drops. This section provides practice for calculations for these types of prescriptions.
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.
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).
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.
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.
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:
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.
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.
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.
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.
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.
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.
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)
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?